IN7ERPRETATIOW OF WELL TESTS In ACUTE FRACTURE … · 2004-11-29 · iii The problem of well test...
Transcript of IN7ERPRETATIOW OF WELL TESTS In ACUTE FRACTURE … · 2004-11-29 · iii The problem of well test...
IN7ERPRETATIOW OF WELL TESTS In ACUTE FRACTURE-YELLBORE SYSTEMS
A thesis submitted to the
School o f Graduate Studies
in partial fulfilment o f the
requirements for the degree o f
Doctor o f Philosophy
Department of Earth Sciences
Memorial University o f Newfoundland
August 1994
St John's Newfoundland
~auonal u~rary DIUIIUU 1-u~ I ~ U U I WIG 191 of Canada du Canada
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FRMISPIECE: Views o f the mdel installed i n the box frame (top) and the
experimental set-up (bottom).
i i i
The problem of well test interpretation in acute systems has been
investigated both theoretically and experimentally. This investigation :
a) establ ishes a basic understanding of the near we1 lbore flow mechanism
in acute systems; b) formalizes the intersection angle-dependent
variations in the streamline pattern and hence in pressure distribution
and observed response; and c) provides mathematical tools to predict these
variations.
The theoretical component of this study involves: a) the derivation,
for acute systems, of the governing differential equation of flow in
fractures; b) the introduction of analytical models for constant-flux
tests under transient and steady-state conditions; c) the formulation of
the stream1 ine-equipotent ial network created by inject ion/pumping through
acute systems under initially non-uniform heads; and d) the development of
a general, semi-analytical model accounting for the roughness, turbulence
and intersection effects in interpreting single-well, constant-flux tests.
The experimental investigation is intended: a) to verify the
formation of the idealized streamline pattern and examine the effects of
likely interactions at the acute intersections, particularly during
injection tests; and b ) to quantify the exit/entry loss coefficients as a
function of the intersection angle. The experimental set-up designed to
carry out this investigation includes three distinct fracture-wellbore
system models w i t h goa, 20. and 10' intersection angles. The laboratory
iv
programme involved testing these models for three different apertures
under steady, constant-flux, injection and pumping conditions. The overall
experimental set-up successfully simulated the conceptual testing
environment which the mathematical model is expected to reproduce.
The pumping pressure distribution observed in the acute system model
tests is in good agreement with the predictions whereas the injection
pressure distribution at large intersection angles proves to be
directionally variable. Although the latter poses a theoretical
limitation, the mathematical models, in practice, are equally capable of
interpreting single-well, injection as well as pumping test data, and are
valid for the design o f the wellbore activities.
I would like to thank Dr. J . Malpas for providing me a shelter under
which I experienced the joy of academic independence and the pain of self -
supervision. Without his positive attitude, I would have little reason to
celebrate. I would like to express sincere appreciation to Drs. 6.
Quinlan, J. Malpas and W. Jamison for reading the manuscript and helpful
comments . I gratefully acknowledge the financial support by the School o f
Graduate Studies, the Department of Earth Sciences, the Faculty o f
Science, and the NSERC grant to Dr. J. Malpas.
Tony Randell of the NRC Institute of Marine Dynamics kindly provided
free time and technical assistance for the use of the computerized lathe.
Doug Boulger of Engineering Technical Services of the Memorial University
ably assisted during the preparation of the experimental set-up.
I would like to thank my wife, Nurdan Aydln, for consistently
sharing the excitements and troubles of our student lives at home and
abroad.
This thesis is dedicated to my dear parents, Miinevver and Necmettin
A y d m .
TABLE OF CONTENTS
Page
FRONTISPIECE . . . . . . . . . . . . . . . . . . a . . . . . . . i i
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i i
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . v
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES AND TABLES . . . . . . . . . . . . . . . . . . . x
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . a . . . . . - xiv
GLOSSARY OF SELECTED TERMS . . . . . . . . . . . . . . . . . . . xvi i i
1 INTRODUCTION . . . . . . - . . . . . . . . . . . . . . . . . . . 1
1 .1S ta te rnen to f theprob lem . . . . . . . . . . . . . . . . 1
1.2 Purpose and scope . . . . . . . . . . . . . . . . . . . . 3
2 FUNDAMENTALS OF FLOW I N FRACTURED MEDIA . . . . . . . . . . . . . 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Theoretical basis o f single-fracture models . . . . . . . 6
2.3 Deviations f r o m Poiseuille law . . . . . . . . . . . . . 9
2.4 Hydraulics of parallel plate conduits . . . . . . . . . . 11
2.5 Flow in rockmasses . . . . . . . . . . . . . . . . . . . 17
2.5.1 Directional permeability o f equivalent continua . 17
2.5.2 Heterogeneity i n fractured rocks . . . . . . . . 21
2.5.3 Statistical models . . . . . . . . . . . . . . . 23
vi i
. . . 3 THEORY AND PROBLEMS OF SINGLE-WELL TESTS IN FRACTURED MEDIA 28
. . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 28
. . . . . . . . . . . . . . . 3.2 Well test evaluation models 31
. . . . . . . . . . . 3.3 Basic models o f constant-flux tests 31
. . . . . . . . . . . . . . . . . 3.3.1 Transient head 32
3.3.2Steadyhead . . . . . . . . . . . . . . . . . . . 34
3.4 Nature of measurements in active wells and we17 losses . 35
. . . . . . . . . . . . . 3.5 Single-well constant-flux tests 39
. . . . . . . 3.5.1 Pressure build-up or recovery tests 40
. . . . . . . . . . . . . . . 3.5.2Step.drawdowntests 41
. . . . . . . . . 3.5.3 Geotechnical permeabilitytests 43
. . . . . . . . . . . . . . 3.6 Low-permeability media tests 45
. . . . . . . . . . 3.6.1 Constant-head injection tests 46
. . . . . . . . . . . . . . 3.6.2 Slug and pulse tests 47
4 MATHEMATICAL MODELS FOR WELL TESTS IN ACUTE FRACTURE-WELLBORE
. . . . . . . . . . . . . . . . . . . . . . . . . . SYSTEMS.
. . . . . . . . . . . . . . . . . . . . . . 4.1Introduction
. . . . . . . 4.2 Previous studies relating to acute systems
. . . . . . . . . . . . . . . . . . . . . . . 4.2 Methodology
. . . . . . . . . 4.3 Mathematical formulation of the problem
. . . . . . . . 4.4 Analytical models o f constant-flux tests
. . . . . . . . . . . . . . . . . 4.4-1 Transient head
. . . . . . . . . . . . . . . . . . . 4.4.2 Steady head
. . . . 4.5 Total drawdown in steady.state, two-regime flows
vi i i
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . 68
5 LABORATORY STUDY OF FLOW THROUGH ACUTE
SYSTEMS . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . 5.2 Physical models o f orthogonal systems
5.3 Theoretical basis for model design . 5.4 Description of experimental set-up .
5.4.1 Design of acute system models
. . . . . . . 5.4.2 Steel box frame
5.4.3 Fabrication procedure . . . . 5.4.4 Water circulation system . .
. . . . . . . 5.4.5 Instrumentation
5.5 Experimental design and test results
FRACTURE-WELLBORE
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
6ANALYSISANDDISCUSSIONOF TESTRESULTS . . . . . . . . . . . 95 6.1Introduction . . . . . . . . . . . . . . . . . . . . 95
6.2Background . . . . . . . . . . . . . . . . . . . . . . . 95
6.3 Pumping tests . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Injection tests . . . . . . . . . . . . . . . . . . 102 6.5Sumary . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . 121
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
APPENDIX A, AN EXPRESSION FOR THE PERIMETER OF AN EQUIPOTENTIAL
SURFACE I N A CONVERGEKT/DIVERGENT FLOW FIELD COMPOSED OF
STRAIGHT STREAMLINES NORMAL TO THE ELLIPTICAL INNER
BOUNDARY . . . . . . . . . . . . . . . . . . . . . . . . . 134
APPENDIX B. IN-FRACTURE PRESSURE HEAD MEASUREMENTS . . , . , . . 139
LIST OF f IGURES AND TABLES
Figure Page
2.1 a) An arbitrarily oriented parallel plate conduit and the flow
velocity profile in local coordinates, b ) parameters
characterizing the geometry of rough para1 lel plate conduits. 25
2.2 Fracture flow domains and corresponding friction factors (Table2.1) . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1 Radial half-section through the cone of depression developed
upon pumping from a confined aquifer/fracture under the Theisflhiem assumptions. Note the actual (A) and theoretical (T) head profiles, the components of the total head loss, and the corresponding flow domains. . . . . . . . . . . . . . . . 51
4.1 a) Parallel, orthogonal and acute fracture-wellbore
intersections along a vertical wellbore; b ) corresponding intersection outlines (linear, circular, elliptical) and
streamline patterns (parallel, radial, general); and c)
sketches of total head prof i les upon inject ion under in1 t ial ly uniform head and laminar flow conditions. . . . . . . . . . . 72
4.2 Acute intersection and the Theis type curves simulating the transient response at the active well. . . . . . . . . . . . 73
4.3 The zone of influence upon injection into acute and orthogonal
systems with inclined fractures in unsaturated zones. . . . . 74
4.4 Comparison o f aperture predictions for given total heads at the wellbore face: steady, two-regime flow through rough
fractures. . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 A generalized view of the experimental set-up with the frame- 20' model assembly placed in the model tank. Water circulation in pumping tests is shown schematically. . . . . . . . . . . 89
5.2 Rate of change of intake/outlet area as a function of the intersection angle. Figures are normalized with respect to the
area of orthogonal intersections with equal diameter . . . . . . . . . . . . . . . . . . . . . . . . . wellbores, 90
5.3 a) Pressure hole locations with respect to the fracture mid- planes; b ) 3-D view o f the 10' drill hole (only wellbore pressure holes are shown) and c) magnified cross-sections
showing details of 20' model (x 3.0) and of pressure hole construction with tube adopter installed (x 5.0) (INSET). Note the enlargement of the wellbore sections to accomnodate the
. . . . . . . . . . . . . . . . . . . . . PVCpipeextension. 91
5.4 a) The steel box frame and details of frame-model connection
(INSET); and b) plan view of the lower half frame with the model boundaries superimposed. . . . . . . . . . . . . . . . 92
6.1 Profiles o f bounding streamlines of f l o w from a large tank into a pipe mounted: a) normal; and b) oblique to the tank surface (schematized from JSME, 1988). . . . . . . . . . . . 106
6.2 Entry loss coefficients of f l o w from an infinite space into
conduits having different cross-sections and intersecting at different angles. Note the reverse trend in the variation of
the loss coefficients as the ratio v,/v increases (Inset).
Prepared from an empirical expression (solid line) and data (dashed lines) reported in Fried and 'Idelchik (1989). . . . . 107
6.3 a) Idealized model of flow in a co-axial enlargement; and b )
discharge from tubes of varying cross-sections mounted on a wall with passing stream (after Fried and Idelchik, 1989). .
6.4 Pumping pressure head vs. logarithm o f equivalent radius:
f3=90°; 2b=l.l w; Run no. 4 (Table 5.1). . . . . . . . . . .
6.5 Pumping pressure head vs. logarithm of equivalent radius: 6=20° ; 2b=l. 1 mn; Run no. 6 (Table 5.1). . . . . . . . . . .
6.6 Pumping pressure head vs. logarithm of equivalent radius:
f3=10°; 2b=l.lnm; Runno. 8 (Table5.1). . . . . . . . . . .
6.7 Pumping pressure head vs. logarithm of equivalent radius: B=~o' ; 2b=0 .6 m; Run no. 3 (Table 5.1). . . . . . . . . . .
6.8 Pumping pressure head vs. logarithm of equivalent radius:
6=20a; 2b=0.6 nm; Run no. 1 (Table 5.1). . . . . . . . . . .
6.9 Pumping pressure head vs. logarithm o f equivalent radius:
6=10°; 2b-0.6 nm; Run no. 3 (Table 5.1). . . . .
6.10 Injection pressure head vs. logarithm of equiva
B=90°; 2b-1.1 mm; Run no. 6 (Table 5.1). . . . . lent radius:
lent radius: 6.11 Injection pressure head vs. logarithm of equiva B=20'; 2b=l.l m; Run no. 5 (Table 5.1). . . . .
6.12 Injection pressure head vs. logarithm of equivalent radius:
6=10°; 2b=l.lnm; Runno. 4 (Table5.1). . . . . . . . . . .
6.13 Injection pressure head vs. logarithm of equivalent radius:
f3=90°; 2b-1.6m; Runno. 3 (Table5.1). . . . . . . . . . .
6.14 Injection pressure head vs. logarithm of equivalent radius:
xiii
g=20°; 2b=1.6 nm; Run no. 1 (Table 5.1). . . . . . . . . . . 119
6.15 Injection pressure head vs. logarithm of equivalent radius:
f3=10° ; 2b=1.6 m; Run no. 6 (Table 5.1). . . . . . . . . . . 120
A.1 Pictorial definition of the terms used in the derivation. . . 138
B.1 Location map of the in-fracture pressure measurement holes. . 140
Table
2.1 Friction factors governing fracture flow in the domains
delineated in Figure 2.1 (after Louis, 1974). . . . . . . . . 27
5.1 Outline of experimental data from steady f l o w tests with cormnon set-up parameters for each run series. . . . . . . . . 93
xiv
LIST OF SYMBOLS
parallel plate aperture; aquifer thickness
cross-sectional area of conduit
linear formation loss constant
well loss constant
characteristic length of void structure
hydraulic diameter
fracture scanline vector
conductivity modification factor
pulsed-head well function
skin factor
gravitational acceleration
constant-head well function
hydraulic head
initial hydraulic head
total head
head in tubing or test interval
initial head in tubing
components o f overall head gradient
(zero order) modified Bessel function
J, (zero order) Bessel function
J, (first order) Bessel function
k' permeability
k& permeab i 1 i ty tensor
k mean absolute height of protrusions
K hydraulic conductivity
& (zero order) modified Bessel function
K, (first order) modified Bessel function
1 distance from and normal to wellbore face
1, critical distance
1, distance to outer boundary
L distance to observation well; entry length
ni,nj direction cosines
N permeability shape factor for void geometry
P hydrostatic pressure
q flow rate per unit width of parallel plate conduit
Q flow rate
r, distance to boundary
r, critical radius; radius o f open tubing
ri radius of influence
r, radius of wellbore in test interval
r, wellbore radius
R radial distance to observation well
Re Reynolds number
s drawdown/rise i n hydraulic or total head
s, exit/entry head loss
s, linear head loss
s, nonlinear head loss
s, wellbore head loss
s' res idua l head - s Laplace transform variable
s storat i v i t y
S, specific storativity
t time
t, time at which boundary is f e l t
shut-in time
transmissivity
unit vector i n mean flow direction
(microscopic) velocity
average (macroscopic) velocity
average velocity normal to equipotential surfaces
velocity o f passing stream; wellbore velocity
exit velocity
V, initial fluid volume in test Interval
w width o f conduit
xvi i
Y, (zero order) Bessel function ---
Y, (first order) Bessel function ---
w constant-flux we11 function - - -
z elevation head L
a kinetic energy corr. factor; complementary position angle ---
fracture inclination from horizontal
fracture-wellbore intersection angle
Euler's constant
wetted perimeter; perimeter of equipotential surfaces
Kronecker delta
perimeter of ellipse divided by 271
acute intersection well function
hydraulic conductivity under turbulent flow conditions
position angle of a point on an elliptical curve
friction factor
kinematic viscosity
exit/entry loss coefficient
entry loss coefficient
we7lbore loss coefficient
exit loss coefficient
density of fluid
compressibility of fluid
compressibility of solid matrix
rp porosity
xviii
GLOSSARY OF SELECTED TERMS
Active well: a well in which the pressure disturbance is induced.
Entry loss: head loss associated with entry of fluid into the fracture.
Exi t loss: head loss associated with exit of fluid from the fracture.
Fabrication aperture: distance adjusted between the model plates during the model fabrication.
Fracture: natural and man-induced rock mass discontinuities whose third
dimension ( i . e . aperture) is much smaller than their areal dimensions. The natural discontinuities range from faults and joints to bedding and schistosity. As their mode of occurrence varies, their dimensions, volumetric density, orientation distribution,
aperture and roughness characteristics exhibit great diversity.
Fully established flow: part of flow in which velocity profile is independent of the entry effects.
Head: mechanical energy per unit weight of fluid.
Hydraulically m o t h : flow behaviour of a fracture whose asperities do not
disturb the boundary layer structure at a given Reynolds number.
Linear flow: laminar convergent/divergent f l o w characterized by linear
total head distribution along streamlines.
One-dimnsional flow: laminar/turbulent flow parallel to an axis of local
cartesian coordinates. Also called rectilinear, parallel or unidirectional flow.
Sfngle-well tests: well tests where the response of the subsurface medium
is observed in the active well.
Wells provide access to the subsurface for purposes such as: a)
resource exploration, exploitation and management; b ) geotechnical site
assessment, remediation and monitoring; and c) waste disposal and
containment. Depending on the nature and operating environment o f these
projects, well testing is needed at various stages to assess such
parameters as: a) the extent and producibi 1 ity of the reserve; b ) the cost
of production, drainage or artificial recharge; and c) the stability and
tightness o f the ground.
Well testing is based on the principle of: a) inducing a pressure
disturbance; b ) monitoring the flow/pressure response of the perturbed
medium; and c) drawing inferences about the hydraulic properties of the
medium. The corresponding procedure is: a) to remove/add fluid or stop
production; b) to record transient/steady pressure (and flow rate); and c)
to select an appropriate mathematical model that can reproduce the
observed response. This way of identifying the properties of an
unknown/incompletely defined system (i.e. lnverse problem) inevitably
results in non-unique answers. Being based on various simplifications,
none o f the models may be assured of reproducing the behaviour o f the
medium under different levels and/or periods of disturbances. The better
the subsurface control and wider the range of tests, the more evolved and
reliable the selected model is for designing planning strategies.
1.1 Stateaent o f t h e problem
In the evaluation o f well tests the observed relationship between
2
the disturbance and the response is attributed to the nature of the system
as conceptualized. In other words the accuracy of the model predictions
remain unknown since the influence of any overlooked system parameter on
the response is implicitly absorbed by the lumped model parameters such as
hydraulic conductivity. In this respect, fundamental investigations toward
a better understanding of the role of every identifiable system parameter
are essential for the progress of well test evaluation. The investigation
of the influence of the fracture-wellbore intersection angle is therefore
a step forward in this direction. A particular gain of this is in refining
various predictions made through single-well tests.
The probability that a wellbore intersects natural fractures at
acute angles is close to certainty. To a lesser extent, this is also true
for hydraulic fractures since the least principal stress may not be
aligned exactly parallel or normal to a plane including the wellbore axis.
However, because of the absence of relevant models for well-test
evaluation, intersect ions of fracture-we1 lbore systems are usual ly
approximated as being parallel or normal to such a plane. How much error
i s involved in this and which approximation is best for a given acute
intersection cannot be answered definitively.
In some cases, the system intersection angle required by the
employed model is produced via oriented drilling. In geotechnical
applications, for example, it is an established practice to drill normal
to the fracture (set) whose permeability is to be determined (Louis and
Maini, 1970; Rissler, 1978). The possibility of making positive
permeability determinations in acute systems so that the investigated rock
3
volume can be sampled more homogeneously with a given number of wellbores
is worthwhile to pursue. Again for interpretation purposes, Hot Dry Rock
(HDR) geothermal recovery wells (deviated at the production levels) are
assumed to orthogonally intersect the hydraulic fractures created in the
injection wells (Murphy, 1979). For fluid recovery wells, in general, the
impact of producing such an intersection on the well losses and hence on
the product ion cost cannot be assessed accurately.
1.2 Purpose and scope
The objectives of this study are therefore: a) to establish an
understanding of the influence of the (fracture-wellbore) system
intersection angle on the near well flow mechanism; and b) to develop
mathematical models which can reproduce the responses of acute systems to
various forms o f disturbances under different conditions. The pursuit of
these interlacing objectives involves, as a base: a) a synthesis of the
present knowledge of flow in fractures and fracture-we1 lbore systems ; and
b) derivation of a basic differential equation governing flow in
conceptualized acute systems. Thereafter, the study focuses on: a) seeking
analytical solutions to this equation for transient and steady-state
constant-flux test conditions; b) investigating the extent and geometry of
the zone influenced by the induced disturbances; c) extending the steady-
state solution to a semi-analytical model accounting for the effects of
fracture roughness, flow turbulence and system intersection on single-well
test results; and d) devising physical models to conduct an experimental
study of actual flow processes in acute systems as conceptualized.
2 FUWDMNTALS OF FLW I M FRACTURED MEDIA
It is evident that the influence of the system intersection angle on
the well test response can be investigated in a general sense only if the
system components can be idealized to some comon forms, e.g. a hollow
cylinder for the wellbore and a parallel plate conduit for the fracture.
It is then essential to understand the mechanics of flow through such
forms and the theoretical concerns behind such idealizations. Because the
results of well tests are often extrapolated to a larger scale, it is also
important to analyze the nature of the rock-mass scale heterogeneities and
modelling methods. These fundamental points are reviewed in a critical and
comprehensive manner in the following sections.
2.1 Introduction
The Navier-Stokes equation, the general equation of Newtonian fluid
motion, can be derived from simultaneous consideration of forces acting on
an infinitesimal fluid element (Rouse, 1961). For isothermal flow of an
incompressible fluid, considering that the body forces usually consist
only of the weight forces, the Navier-Stokes equation may be written, in
local cartesian coordinates, as
where v: (microscopic) velocity,
t: time,
p : density,
g: gravitational acceleration,
P: hydrostatic pressure,
z: elevation from a datum, and
v : kinematic viscosity.
The fundamental flow laws that constitute the core of the fracture flow
theory are based on this equation. Integrating Equation 2.1 over a
macroscopic (representative) volume o f granular porous media, Hubbert
(1956) showed t h a t the empirical Darcy law of laminar flow is o f a
universal nature at a proper scale. Accordingly, the Darcy law for any
porous medium, - -
- where V: average (macroscopic) velocity,
K: (homogeneous and isotropic) hydraulic conductivity, and
h: hydraulic head, a combined expression o f pressure and
+ z . elevation heads, i.e. - P 9
The hydraulic conductivity term in the Darcy law was shown to be a
lumped, variable parameter (Hubbert, 1940;1956)
K = k 4 v (2 .3)
where k* represents the permeability o f the medium and the role of fluid
is expressed by its kinematic viscosity. The inherent difficulty of
relating the perrneabi 1 ity to some quantifiable property of the medium
6
resulted in theoretical approaches that employ different conceptual models
and medium parameters (Bear, 1972) according to the type of effective
porosity (i .e. granular, karstic or fracture). Section 2.2 describes the
application of such an approach to single fractures using a parallel plate
ideal ization.
Theoretical ly-derived permeabi 1 ity relations hold provided that the
geometry of the real flow domain is we1 1 represented by the model and flow
obeys the Darcy law. Cases where these conditions fail in fracture flow
are discussed in Section 2.3 whereas Section 2.4 explains how these
situations can be treated using empirical modification factors and
completes the formulation at the scale o f a single fracture. The basic
concepts, problems and methodologies of the fracture flow studies at the
rock-mass scale are reviewed in Section 2.5.
2.2 Theoretical basis o f single-fracture models
Rock fractures have often been idealized as planar conduits (Figure
2 . l . a ) bounded by two parallel plates with smooth walls (Baker, 1955;
Huitt, 1956; Snow, 1969; Louis, 1974; Iwai, 1976). This geometric
simplification allows an exact analytical solution of the Navier-Stokes
equation (Equation 2.1) subject to the following conditions (Schlichting,
1979) :
a% - = o : steady flow, at
v ,=v ,=o: one-dimensional flow, in the x, direction,
3 =o: continuity condition (in fully established f l o w through a ax,
constant cross-section), and
3 = O : constant cross-section along the width.
Substituting these, Equation 2.1 reduces to
Writing this in terms o f hydraulic head,
and integrating twice with respect to x, ,
2 vl=32!L2+clx3+q v ax, 2
where C, and c, are the integration constants.
The no-slip condition at the boundaries, i . e . V . = O at x,=*b,
requires that
where b i s the half-width of the parallel plate opening (Figure 1). Thus
- CJ ah d-b2 vl-- - v ax, 2
which shows that the velocity profile is parabolic under the specified
flow conditions.
Averaging the velocity across the flow section
+b
yields a linear relationship known as the Poiseuille law
- =- g(2b) ah V1
- 1 2 v ax,
which is a specialized form of the Darcy law (Equation 2.2). The hydraulic
conductivity o f a fracture with smooth, parallel walls therefore is
It follows from Equation 2.3 that the permeability of this idealized
fracture is given by
1 k;=- ( ~ b ) ~ 12 (2.12)
which compares to the typical permeability expression suggested f o r
granular porous media (Hubbert, 1940)
k* =N@ (2.13)
where N is a shape factor depending on certain properties describing the
9
void space geometry of the specific medium (e.g. angularity of grains) and
d is a characteristic length of that medium (e.g. mean grain diameter).
From a visual inspection of Equations 2.12 and 2.13, the shape factor for
the parallel plate flow geometry appears to be
It is of both practical and academic interest to determine when Equation
2.14 becomes inadequate in describing flow geometry of rock fractures.
The Poiseuille law (Equation 2.10) can be expressed in terms of q,
the f l o w rate per unit width of the fracture,
4= - g(2b13 - ah 1 2 v ax,
which is the so-called cubic law. The Poiseuille (or cubic) law may be
taken as approximately valid for parallel plate conduits having gradual
variations in the local apertures and/or curved (undulating) surfaces, the
radii of curvature of which are large compared with the local apertures
(Lamb, 1945). In both forms (Equations 2.10 and 2.15) the sunarized
solution of the Navier-Stokes equation has been the basic model in the
study of laminar flow through rock fractures.
2-3 Deviations from Poiseuille law
Fracture flows that can be ful ly characterized by the Poiseui 1 le law
(Equation 2.10) are rare occurrences in nature. The deviations stem from:
a) oversimplifying the fracture void structure; and b) ignoring the
inertial forces. Fractures generally are planar features with surfaces
10
that can be characterized by asperities superimposed on larger scale
undulations, These surface elements often produce a highly compl icated
void structure surrounding and enclosed by areas in contact, and hence
tortuous flow paths throughout the fracture plane (Sharp and Maini, 1972;
Pyrak-Nolte et al. 1987). As the fracture void structure becomes
increasingly different from a simple opening between parallel smooth
walls, the theoretical permeability (Equation 2.12) cannot sustain the
1 ineari ty between mean flow velocity and corresponding gradient, although
f l o w may still be of laminar character.
Groundwater flow usually occurs under low hydraulic gradients (Bear,
1979). However, in many engineering environments, such as those near
active we1 lbores and excavated faces, art if icial disturbances frequently
induce steep gradients. The consequent increase in inertial forces is
accompanied by shifting of streamlines due to flow separation at the
diverging and/or curved points of the void structure (Bear, 1972). This
implies that the microscopic structure of the effective voids imposes an
additional control on the initiation and intensity of flow separation.
Increasing significance of inertial forces relative to shear forces
results in a nonlinear relationship between mean velocity and hydraulic
gradient, regardless of the void structure.
In summary, deviations from the Poiseuille law can be attributed to
tortuosity and/or inertial forces. From a practical point o f view, the
deterministic approach is ideal for the treatment of these compl icat ions.
The main concern in this approach is not in the details of the void
structure ( i . e . in geometric similarity between the model and fractures),
11
but in reproducing equivalent responses to given excitations. In selecting
the model it is advantageous: a) to maintain the theoretical base gained
by the Poiseuille law; and b) to use the parallel plate aperture as a
parameter providing physical continuity for the entire formulation.
In accordance with the above rationale, major experiments designed
to investigate hydraulic behaviour of fractures (Baker, 1955; Huitt, 1956;
Parrish, 1963; Louis, 1974, Rissler, 1978; Cornwell and Murphy, 1985) have
consistently used physical models consisting of parallel plate conduits of
varying roughness (Figure 2.l.b). Although introducing roughness as an
additional parameter improves void structure characterization, it does not
entirely prevent some loss o f control over the phenomenon being
investigated. Owing to the large number of and complex relations between
the (unknown) system parameters involved in well test evaluations and
rock-mass scale characterization, present practice relies heavily on
developments based on the parallel plate idealization. The next section
explains the procedure that enables the use of the parallel plate concept
in establ i shing flow express ions for rough fractures and/or nonl inear
conditions for which the Poiseuille law fails.
2.4 Hydraulics o f parallel plate conduits
For fully established flow through straight, uniform conduits, the
difference in the hydraulic head between any two points arises from the
energy dissipation due to both: a) viscous shear and/or turbulent mixing
within the fluid body; and b) frictional and/or pressure drag on the
conduit walls and protrusions. The resulting head gradient, regardless of
12
the causes, can be predicted from the Darcy-Weisbach equation, originally
derived for pipes of circular section (Vennard and Street, 1982). Dropping
indicia1 and partial derivative notations, this empirical equation has the
general form
where A : friction factor, and
Dh: hydraulic diameter.
The conduit geometry in Equation 2.16 is specified through the
hydraulic diameter term defined as
where A: cross-sectional area, and
r: perimeter (i-e. length of contact between fluid and
boundary).
Hence, for an opening between two parallel plates,
where w is the width of the opening.
The Darcy-Weisbach equation (Equation 2.16) does not explicitly
account f o r the influences of roughness, flow regime and fluid viscosity.
The friction factor can, however, be effectively scaled by means of two
dimensionless parameters (Vennard and Street, 1982): a) relative
roughness, the ratio o f mean absolute height o f the protrusions, k (Figure
13
2.l.b) to the hydraulic diameter, D, (Equation 2.18); and b) Reynolds
number, the ratio of inertial to viscous forces, and therefore an index of
the flow regime,
On the basis of the hydraulic approach outlined above, several
investigators including Huitt (1956) and Louis (1974) have attempted to
empirical ly determine friction factor express ions from rough model
fractures under a wide range of flow conditions. The friction factor
expressions adopted in this study are listed in Table 2.1. The domains of
each expression are delineated in Figure 2.2. Before explaining the
integration of the expressions into flow laws for these domains, the
following properties of Table 2.1 and Figure 2.2 need emphasis and
clarification:
a) the transitions from; (i) laminar to fully turbulent flow, ( i i )
hydraul ical ly smooth to completely rough behaviour, and ( i i i)
parallel (k /D,sO. 033) to nun-parallel ( k / D , > 0.033 ) wall
geometry are all assumed to be abrupt, resulting in discontinuous
boundaries between the flow domains,
b) above the boundary value o f k/~,=0.033, the friction factor
corresponding to the Poiseuille law is no longer valid due to
additional energy dissipation in the process of viscous damping of
increased wall disturbances,
c) below k/4=0.033, the critical Reynolds number, marking the
persistence limit of laminar flow, is constant at Rec=2300,
d) below k&=o .033 , the friction factors of classical hydraulics
are valid, and
e) below k/D,=0.033, in the turbulent domain, as the viscous
sublayer gets thinner and/or disrupted with increasing Reynolds
number, the resulting exposure of protrusions changes hydraulic
behaviour of the model fracture from smooth to rough.
Lsrinar d o n ins
The friction factor denoted as Poiseuille (Table 2.1) can be derived
from equating hydraul ic gradient terms in the Darcy-Wei sbach (Equation
2.16) and Poiseuille (Equation 2.10) equations and making use o f Equations
2.18 and 2.19. Comparing this with the friction factor o f Louis L
(laminar), the modification factor, f, necessary in the laminar domain
with k / q > 0.033 appears to be
Introducing this as an external parameter into the Poiseuille law
(Equation 2.10) reveals that the shape factor (Equation 2.14) is a
function of surface roughness in this domain
Turbulent d o n i n s
Two distinct forms of flow relationships are widely used to
15
formulate the observed nonlinearity between the velocity and hydraulic
gradient (Basak, 1978; Hannoura and Barends, 1981): a) the series form o f
the Forchheimer equation
and b) the exponential form of the Missbach equation
where a, c, and q are empirical proportionality coefficients. The symbol
q denotes the hydraulic conductivity of a fracture under turbulent flow
conditions. The Forchheirner equation provides a better alternative in
cases where the transitional turbulent flow persists over a considerable
spatial extent for a wide range of flow rates, such as in granular porous
media (Bear, 1972). However, for (particularly convergent/divergent) flow
through fractures, transition to fully turbulent flow is more abrupt.
Therefore, it i s appropriate to apply the Missbach equation in the
turbulent flow domains shown in Figure 2.2 (Louis and Maini, 1970; Louis,
1974; Zeigler, 1976; Rissler, 1978).
In accordance with the assumption o f abrupt transition to fully
turbulent flow (Figure 2.2) , the Missbach equation (Equation 2.23) is
taken to be quadratic, i .e. n =2. Equating the Missbach and Darcy-Weisbach
(Equation 2.16) equations yields the turbulent domain hydraulic
conductivity, q , in terms of the empirical friction factors and the
parallel plate aperture
Crit ica 1 Reyno Ids nuder
Any observed relationship between the mean velocity and hydraulic
gradient can now be formulated by selecting the appropriate flow law
(Equation 2.10 or 2.23) and then adjusting its proportionality constant
(i.e. hydraulic conductivity) adopting an appropriate friction factor
expression (Table 2.1). For this purpose, the values of the relative
roughness o f the fracture and the Reynolds number o f flow can be directly
compared to the flow boundary domains depicted in Figure 2.2 and Table
2.1. The critical Reynolds numbers defining the boundaries of these
domains are calculated in the following manner:
a) for k/~,r0.033: the critical Reynolds number, Re,, is
essentially constant (Figure 2.2)
Re, = 2300
b) for k/~,>0.033: Re, can be obtained by the simultaneous
solution of the friction factor expressions of Louis L (laminar) and
Louis T (turbulent) (Table 2.1)
c) for k/~,s0.033 and Re>2300: the boundary between
hydraulically smooth and completely rough flow domains satisfies the
17
friction factor expressions of both Blasius and Nikuradse
Re,=2.553 ( log- ;;zJ8 Any formulation derived by the procedure as outlined includes two
unknowns, namely, the aperture and hydraulic gradient. In practice,
hydraulic gradient is measured in order to determine the equivalent
parallel plate aperture of fractures.
2.5 Flow in rock masses
Assessment of the seepage, production or contaminant transport
potential in fractured media ideally requires knowledge of: a) the spatial
distribution of active fractures; and b) the geometric, hydraulic and
mechanical properties o f these fractures. Obviously such a thorough
description of the flow network is practically an impossible notion.
Therefore studies on flow through fractured rock masses have adopted
indirect approaches assuming the existence o f : a) an equivalent (granular)
porous medium (EPM) behaviour (Snow, 1969; Castillo, 1972; Louis, 1974);
and b) statistically equivalent networks of discrete fractures (Long and
Witherspoon, 1985; Schwartz and Smith, 1985; Rouleau, 1988; Nordqvist et
al., 1992).
2.5.1 Directional permeability o f equivalent continua
Fractures impart anisotropy and heterogeneityto the permeability of
rock masses. In anisotropic media, the Darcy law (Equation 2.2) becomes
(Bear, 1972)
where k& is a second order tensor defining the permeability at a point
and rj denotes the components of the overall head gradient. This equation
implies a fictitious continuum replacing the multiphase (solid and pore)
medium. Therefore the tensor represents the average permeability of a
certain volume of the actual medium centred at that point. If the
permeability is insensitive to slight changes in this volume, it is
specified as the representative elementary volume (REV) (Hubbert, 1956;
Bear, 1972). In order that the same REV can be defined at all points of
the flow domain (Bear, 1979), heterogeneities should have a high
volumetric density (relative to the REV) which can be uniform or vary
smoothly across the flow domain. Consequently a fractured medium is said
to behave like an EPM (or a continuum) i f the REV exists at a scale
smaller than that of the measurement and also of the detail required in
the studied flow problem (Neuman, 1987).
A method to calculate the directional permeability tensor in
fractured media was developed by Snow (1969) who implicitly assumed the
existence of the REV with uniform heterogeneity density. Incorporating the
cubic law modification factor to the equivalent parallel plate
permeability, the tensor for a single fracture is (Snow, 1969; Rissler,
1978)
- -
where 6,j : Kronecker delta,
ni,nj: direction cosines of the norm1 to the fracture, and
D,: scanline vector o f length D.
The underlying assumptions of the method are that: a) all fractures
traversed repeat for every scanline length D so that the orthogonal
distance (or spacing) between each fracture and its image is In,~,l; b)
fractures are continuous (or extend to a specified boundary); and c) there
is no flow interference at the fracture intersections.
The first assumption implies that the rock is homogeneous at the
scale of the scanline length since identical heterogeneities can be
sampled along every such length. The average permeability calculated from
such a representative elementary length (REL) equals the REV (Bear, 1972).
The summation o f a1 1 single-fracture tensors for each scanl ine stat ion
yields the permeability tensor for the REV. As the number of stations
increases, the tensor is refined by averaging, hence reducing the sampl ing
bias. Also the more extensive the fracture is, the higher its chance of
being traversed by multiple scanlines and being weighted more heavily.
Therefore, for the multiple scanl ine surveys the REL contains an imaginary
but more representative sample of fractures.
The second assumption indicates t h a t contribution of a fracture to
the overall perrneabil ity is not affected by its network connectivity. Each
fracture encountered along a scanline contributes to the permeability of
the REV independently and proportionally to its equivalent aperture,
roughness and orientat ion with respect to the overall gradient vector.
20
Fractures oblique to the scanline are weighted more heavily to eliminate
the orientation bias.
Flow rate reduction and pressure loss at the fracture intersections
due to cross flow (i.e. flow interference) are negligible in laboratory
model experiments in the laminar flow range (Wilson and Witherspoon, 1976)
verifying the third assumption. However, Neuman (1987) suggested,
referring to a field case study in the literature, that intersections may
exert a greater influence on the overall hydraulic conductivity than do
fracture planes.
The magnitude and directions of the principal permeabilities are
obtained from the eigenvalues and eigenvectors, respectively, of the final
permeability tensor. Velocity and gradient vectors in an anisotropic
medium do not coincide except in the principal directions of permeability.
The polar plot of the inverse square root of the permeability in the
direction of the overall gradient yields an ellipsoid whose axes are the
principal directions o f permeability (Bear, 1972). The degree of
elipticity reflects the degree of deviation from a continuum behaviour.
The influence of various network parameters on continuum behaviour
was investigated by several two dimensional network models. These include
the conceptual (square and triangular) grid models for the aperture
variations (Parsons, 1966), the resistivity analogs of a square grid model
for the finite size of fracture sets (Caldwell, l972), and the statistical
networks for the degree of interconnection (Long and Witherspoon, 1985).
The maximum permeability calculated by the outlined method of Snow (1969)
was found to be a reasonable approximation in homogeneous media given that
21
the length of fractures exceeds a certain limit (Long and Witherspoon,
1985).
2.5.2 Heterogeneity i n fractured rocks
The type of medium determines the scale at which the REV may exist.
Fractured rock masses usually exhibit several episodes of fracturing, each
resulting in a higher level of heterogeneity at a given site (Chernyshev
and Dearman, 1991). These levels display differences not only in fracture
density but also in connectivity and ability to form a global flow
network. Networks formed by dense but isolated, or sparse but active
fractures are natural probabilities. Accordingly there may be more than
one volumetric scale at which the REV behaviour exists at a given point
(Wilson et al., 1983; de Marsily, 1985; Smith and Schwartz, 1985). Only
the largest o f these might be the true REV scale for the flow domain.
A number of field observations imply that geometrically and
hydraulically defined fracture frequencies can be very different. From
measurements of injection pressures at which pre-existing fractures start
opening, Cornet (1992) concluded that stress heterogeneities are
associated with active zones, and the regional stress field is not
perturbed by most of the fractures. Similarly, Tsang et al. (lWO), using
a time sequence of electrical conductivity logs, detected merely nine
active fractures scattered along a 900 m wellbore interval. The preceding
observations also indicate that, although there is a general decline in
the well yield with increasing depth at shallow depths of hydrogeological
interest (Wooley, 1982), this is not a rule for long intervals
22
particularly when away from the zone of surface weathering and
percolation.
A parametric study using two dimensional synthetic networks of
randomly distributed and oriented fractures (i.e. homogeneous and
isotropic at the REV scale) demonstrated that the existence and scale o f
the REV are strongly dependent on the length of the fractures (Long and
Witherspoon, 1985). The occurrence of fractures i n clusters is, however,
a comon phenomenon inconsistent with the assumption of randomness (Snow,
1970; de Marsily, 1985; Schwartz and Smith, 1985). To form a global flow
network, fractures in such cases may have to be extremely long for the REV
to exist if local networks are not connected by several pervasive
fractures.
These results emphasize the importance of improving in situ
detection, testing and evaluation methods to obtain more refined estimates
of network parameters and to identify the most significant fractures at a
given scale (Wilson et al. 1983). Particularly, single-hole packer testing
to obtain hydraulically derived geometrical information or direct local
permeability is an essential method (Wilson et al. 1983; Neuman, 1987).
Techniques to determine fracture connectivity include, in accordance with
the nature of the problem, cross-hole tests involving multi-level
measurements of pressure signals (Hsieh, 1987), temperature (Silliman and
Robinson, 1989) and gamma ray (Marine, 1980) variations in response to
injection or pumping.
Hydraulic characterization of fractured rock masses generally
requires the use of multiple sources of information to assess
23
heterogeneity variation at different scales (Hsieh, 1987). In planning to
acquire such information, geological and mechanical controls of fracturing
intensity such as lithology, thickness, structural association and depth
(Stearns and Friedman, 1972) should be considered first. Geophysical and
hydrological testing should follow this guide in delineating areas of
uniform heterogeneity where the permeability tensor can be determined and
in extrapolating the available data to the whole flow domain. The extent
of testing to be undertaken depends on the quality, quantity and variety
of the data needed, which in turn is determined by whether there is REV
behaviour. Statistical models incorporating the available geometrical
fracture data might be useful in answering these questions.
2-5.3 Statistical models
The EPM approach assumes a priori that REV exists, while studies
with discrete fracture networks test whether and under what conditions the
REV might exist. Discrete fracture network models need statistical
information about the geometry and spatial distribution of fractures.
Density, length, location, orientation, aperture, etc. of fractures are
considered random entities from a probability distribution function (Long
and Witherspoon, 1985; Rouleau, 1988). Distribution parameters are
estimated from sample observations. These distributions are then randomly
sampled to generate statistically equivalent networks.
Different levels of heterogeneity can be modelled by including major
fractures separately (Wilson et al., 1983) or by superimposing
independently generated fracture assemblages (Long and Witherspoon, 1985).
24
Single fractures are idealized as, for example, parallel plate discs,
rectangles, etc. whose aperture may be constant (Rouleau, 1988) or
variable (Nordqvist et a1 . , 1992). Distribution o f apertures (Bianchi and
Snow, 1969) as well as lengths (Long and Witherspoon, 1985) of fractures
in a sampled rock volume might obey a lognormal distribution function.
Since flow network parameters such as connectivity and areal extent
cannot be directly measured, statistical network models are designed to
predict them using other parameters such as frequency and trace length.
More significantly there may be a few fractures controlling network
connectivity in which case their apertures cannot be simply estimated. An
alternative to avoid these shortcomings is a method which treats the
conductivity values determined by single-hole packer tests as random
variables generated by a stochastic process defined over a continuum
(Neurnan, 1987).
2 b : Equivalent apertu~e
k : Absolute ~oughness
Figure 2.1.a) An arbitrarily oriented parallel plate conduit and the flow velocity
profile in local coordinates, b) parameters characterizing the geometry of rough
parallel plate conduits.
Table 2.1. Friction factors governing fracture flow in the domains delineated in Figure 2.1 (after Louis, 1974 ) .
I R E U TiVE ROUGHNESS
HYDRA ULlC BEHA VIOR
FRICTION FACTOR
RELA TI VE ROUGHNESS
HYDRA ULIC BEHA VIOR
FRICTION FACTOR
L A M I N A R
SMOOTH
I
(Poiseuille) I I 1
(Louis L)
T U R B U L E N T I
I SMOOTH i COMPLETELY ROUGH
(Blasius) i (Nikuradse) (Louis T)
3 THEORY AND PROBLEMS OF SIN6LE-WELL TESTS I N FRACTURED MEDIA
The intersection angle which determines the shape of the inner
boundary of the fracture flow domain is merely one of the system
parameters that may influence the response observed in the active well. It
is therefore necessary to examine the test situations i n which various
parameters become influential and dominate deviations from the ideal
response. This analysis, in turn, allows the identification of specific
conditions and test methods for which intersection angle can be isolated
as the main parameter for the purposes of the present investigation. These
concerns, relating to the theory and problems of the single-well tests,
are dealt with in this chapter.
3.1 Introduction
The law of conservation of mass requires that the net inward flux
through any arbitrary volume be equal to the rate o f accumulation. For
fluid flow within porous media, this balance (per unit volume and time)
can be expressed as
where cp is the porosity (defined over the REV) and p is the fluid
density. The rate o f accumulation of the fluid mass under confined
conditions is related to the elasticity o f the solid-fluid system
occupying the concerned volume (Jacob, 1940; Bear, 1972)
29
where 0, and a, are the compressibility o f the fluid and the (moving)
solid matrix, respectively, and S, is the specific storativity of the
sys tern.
Since the spatial density variations can be assumed negligible in
most problems (Hantush, 1964), Equation 3.1 when combined with Equation
3.2 reduces to
Assuming laminar f l o w conditions, and hence substituting the average
velocity as determined from the Darcy law for homogeneous isotropic media
(Equation 2.2) yields the fundamental equation of diffusion in porous
media
a2h - S, ah ---- ax: K dt
Here both specific storativity and conductivity are referenced to the
initial values of the system properties. Integrating through the entire
thickness, 2b, o f the saturated zone
where s and T are the storativity and transmissivity of the zone,
respectively,
30
Note that Equation 3.5 applies only to isotropic, non-leaky, confined
aquifer conditions. Variations in this equation address other
possibilities such as anisotropy in permeability, leakage to/from
confining layers, unconfined (gravity) drainage and/or steady-state
behaviour (Hantush, 1964; Kruseman and de Ridder, 1970; Lohman, 1972).
The solution of Equation 3.5 or its various equivalent forms for
given initial/boundary conditions and flow domain geometry allows one: a)
to estimate the subsurface hydraulic properties from field measurements;
and subsequently b) to predict the pressure behaviour upon changes in
boundary conditions or upon natural/artif icial di sturbances . The former is referred to as the inverse problem whereas the latter is the forecasting
problem (Bear, 1979).
The essential aim and usage of well testing is to solve the inverse
problem. Section 3.2 briefly outlines the constitution of the well test
models from this general perspective. The underlying assumptions and the
derivation of the basic constant-flux test solutions are reviewed in
detail in Section 3.3 , This review constitutes the core of the evaluation
models for single-well, constant-flux tests (Section 3.5) and sets a
comparative basis for the mathematical developments presented in Chapter
4. While maintaining the generality in the variety o f test methods, this
chapter thereafter focuses on the basic evaluation models for single-well
tests in confined media. This is to highlight the influence of individual
fracture characteristics on the near well flow phenomena and consequently
on the observed response. First the inherent problems o f the observations
in an active well are thoroughly discussed in Section 3.4. The subsequent
31
sections (Sections 3.5 and 3.6) include brief descriptions of the
application, testing and evaluation procedures for single-well tests and
a critical appraisal of their results with reference to Section 3.4.
3.2 Well test evaluation e l s
The mathematical models used in the analyses of well tests consist
of (analytical) solutions of Equation 3.5 and its variations for different
conceptual models and inner boundary conditions. These conditions vary
according to the test technique (e-g. constant-flux/head, slug, pulse) and
other considerations such as wellbore storage, skin effect, finite well
radius, intersecting fracture, wellbore penetration (Gringarten, 1982;
Karasaki, 1987). The conceptual models are built upon idealizations about
the medium (e-g. confined, nultilayered, composite, double-porosity), the
flow domain geometry (e. g . the outer boundary, thickness, orientat ion) and consequently the f l o w pattern (e-g. parallel, radial, polar). Detailed
general reviews of the analytical models of well test evaluation are
available both for granular porous media (Hantush, 1964; Matthews and
Russell, 1967; Kruseman and de Ridder, 1970; Lohman, 1972) and fractured
media (Zeigler, 1976; Streltsova, 1978; Gringarten, 1982; Karasaki, 1987).
3.3 Basic mdels of constant-f lux tests
The most comnon well tests involve measurements of changes i n
hydraulic head (or pressure) in the observation (and active) wells in
response to fluid suction/injection at a constant rate from/into the
tested medium. Data obtained during such tests conducted in confined media
32
may be evaluated by one of the two fundamental analytical models assuming
that:
a) the medium is homogenous and isotropic, of infinite extent (i-e.
boundary effects are not felt during testing), horizontal and of
constant thickness,
b) the zones immediately overlying and underlying the medium do not
leak under induced vertical pressure differentials,
c) the well fully and vertically penetrates, and uniformly
comnunicates with the medium,
d) the hydraulic head distribution prior to testing is uniform,
e) flow is radially symmetric (as conditioned by a-d),
f) water is instantaneously released from/enters into the storage as
the hydraulic head descends/rises, respectively, and
g) the wellbore radius is infinitely small, i.e. the well acts as a
line sink/source with no self-storage.
The last two assumptions are necessary only when the head response i s
time-dependent (i.e. transient).
3.3 - 1 Transient head
Distribution of transient hydraulic head in a radial flow field can
be expressed in one dimensional form by writing Equation 3.5 in plane
polar coordinates
The initial and boundary conditions posed by the assumptions are
formulated as
where h,: initial hydraulic head,
I,: wellbore radius, and
Q: flow rate.
Re-writing Equation 3.7 in terms of net drawdown/rise in the initial
hydraulic head, i.e. s=ho-h(r,t) , and applying the Laplace transform
method (Hantush, 1964) yields
OD r
where x: integration variable, and
w: constant-flux we71 function.
This solution, first introduced by Theis (1935) by analogy to the
equivalent heat conduction problem, is now known as the Theis equation.
The numerical values o f the function w are obtained from the series
expansion of the exponential integral (Jacob, 1940; Tuma, 1987)
34
where y = O -5772 (Euler's constant).
The Theis equation is extensively used i n practice despite being an
oversimplification. The computation of the two unknowns (S and T) in
Equation 3.9 is accomplished by means of graphical procedures such as
type-curve matching (Theis, 1935) or straight-line plotting through the
so-called logarithmic approximation (Cooper and Jacob, 1946). The latter
is based on the fact that for a long testing period
W(u) = - y -In u
and hence
the semi-logarithmic plots of which form straight lines, regardless ofthe
choice of the independent variable. Among these, the drawdown-time graph
enables the determination of the transmissivity from the measurements in
active wells.
3.3.2 Steady head
Steady-state implies that subsurface pressures have assumed a new
state of equilibrium in response to the induced disturbance. As the
drawdown is a continuous function of time (Equation 3.9), equilibrium is
theoretically impossible. However, when the hydraulic gradient stabilizes
ah ah =,) and/or transients are negligible (5.e. a h / d t = o ) * * ~ 1 1 - 3 E 1 2
after some period of pumping/injection, it is justifiable to assume that
steady-state is attained (Lohman, 1972).
For a h / d t = o , Equation 3.7 reduces to the Laplace equation
When integrated using the underlying boundary conditions
h ( R ) =h,
the result is known as the Thiem equation (Lohman, 1972)
where R is the radial distance to the observation we1 1. In the case o f
single-well data the initial hydraulic head is utilized as an estimate of
the undisturbed head at a point outside the zone of influence.
3.4 Nature o f measurements i n active wells and well losses
Pumping from porous media, under the assumptions of the Theisflhiem
equations (Equations 3.9 and 3. Is), results in an axisymmetric cone of
depression representing the hydraul ic head distribution within the zone of
disturbance (Figure 3.1). According to Equation 3.15 the hydraulic heads
measured simultaneously at any two points along a radial section in this
zone should vary linearly as a function o f the logarithmic distance
between these points. In practice, however, the heads measured in and
around the vicinity o f active wells are substantially lower than
theoretically predicted (Figure 3.1).
36
As the fluid under the induced gradient moves from the radius of
inf hence, through the porous medium, into the we1 lbore and up to the pump
(or tubing) intake, part of i t s mechanical energy is lost to maintain
motion. This loss manifests itself as the difference in hydraulic heads
measured in the active well before and during pumping, i .e. the total
drawdown (Jacob, 1947). It is recognized that the total drawdown consists
of four distinct components (Rorabough, 1953; Bruin and Hudson, 1955)
which in the spatial order of their contribution are (Figure 3.1)
s=sl+s,+s,+s, (3.16)
where s, and s,: linear and nonlinear head losses due to flow resistance
of the medium under laminar and turbulent conditions, respectively,
s,: exit head loss due to sudden enlargement of section and change
in f l o w direction, and
s,: wellbore head loss due to the wall friction within the well.
The preceding discussion equally applies to injection for which: a) the
sequence of the head losses are reversed due to upconing; and b ) the
symbol s, represents the entry head loss due to sudden contraction and
directional change o f flow section.
The first operational relationship between the total drawdown and
pumping rate was introduced by Jacob (1947),
where B and C: linear formation loss and well loss constants,
respectively, and
ri: radius of influence.
The quadratic exponent in the well loss component originates from the
fluid mechanics approximations of the exit/entry and frictional losses in
pipes (Vennard and Street, 1982). The linear formation loss is as given by
the Theisflhiem equations (Equations 3.9 and 3.15).
Rorabough (1953) recognized that above a critical pumping rate,
turbulent conditions dominate flow from a critical radius, r,, to the well
face (Figure 3.1). As the rate increases the critical radius extends
further away implying increasing contribution of the nonlinear formation
loss. Consequently B and c are not constants in practice. However, to
enable the graphical determination of these, Rorabough (1953) modified
Equation 3.17 as
where the exponent n accounts for variations from reference values of B
and C. These expressions (Equations 3.17 and 3.18) constitute the basis
for the analysis of step-drawdown tests.
Measurements in an active well are also influenced by the zone of
altered permeability inmediately surrounding the wellbore. In the case of
enhanced permeabi 1 ity (such as by gravel packing around water we1 ls), the
resulting head recovery i s accounted for by adopting an effective we1 1
radius (Jacob, 1947). On the contrary, a zone of reduced permeability,
38
known as skin (van Everdingen, 1953; Matthews and Russell, 1967) around
oil wells, increases flow resistance. The corresponding head loss is
included in the well loss component of the total drawdown as (Ramey, 1982)
where F, is the skin factor and D, is the constant reflecting a combined
influence o f the nonlinear formation and exit losses.
Flow rate efficiency of a fracture (or fractures of a set) under the
same gradient can vary significantly depending on the distribution of the
effective flow area relative to the active we1 lbore intersect ion (Sharp
and Maini , 1972). This argument was verified by an electrical analog study
(Sundaram and Frink, 1983) and numerical simulations (Smith et al. 1987)
of radial fracture flow. Such pronounced influence o f the near well
fracture geometry occurs because a large percentage of the pumping/
injection head is lost within a short distance o f the wellbore. Therefore
analyses of active well data assuming homogeneity at the scale of the
narrow sampling window of the wellbore intersection may produce highly
biased estimates of hydraulic properties. Observation wells offer the
advantage of avoiding this bias as well as well losses, and of studying
the inter-well connectivity. Aquifer/reservoir scale hydraulic
characterization based on multiple observation wells i s more dependable in
strongly heterogeneous media in which the flow pattern may substantially
differ from the assumed one.
39
3.5 Single-well constant-flux tests
As explained in Section 3.4, head measurements in active wells
during constant-flux tests are altered by an additional, constant amount.
The slope of the data plot (s vs. log t) from the late (infinite-acting)
period therefore yields a transmissivity estimate using Equation 3.12.
This estimate, although free of the effects of well losses and near-the-
well heterogeneities, pertains to the part of the reservoir controlling
the rate of head changes during the late period. This part corresponds to:
a) the matrix domain for the equivalent homogeneous porous medium models
(EHM) in which overall fracture anisotropy is represented by a single,
high-permeability, vertical/horizontal fracture intersecting the active
well (Gringarten, 1982); b) the outer continuum domain for the composite
models (CM) in which inner (concentric) domain consists of discrete
(vertical/horizontal/inclined) fractures intersecting the active well
(Karasaki, 1987); and c) the entire domain within the zone o f influence
for the double-porosity models (DPM) of uniformly fractured porous media
(Streltsova, 1978).
On the other hand, constant-flux test solutions based on the
conceptual models such as EHM and CM predict that the early period is
dominated by the fractures intersecting the active well (if wellbore
storage and skin is negligible). This period, however, lasts only a few
minutes in fractured water wells whereas in oil wells it usually is in the
range of a few hours (Gringarten, 1982). Therefore it may not always be
possible to capture data during this informative period.
In summary, apart from determining the magnitude of well losses and
40
o f near-the-well bias in head measurements in active wells, the hydraulic
characteristics and attitude of these fractures also modify the response
patterns during the early period of testing. Recognition o f these
relations enables the assessment and improvement of the quality of the
inferences made particularly through single-well testing. The constant-
flux test methods usual ly conducted by means of a single we1 1 are examined
below from this standpoint.
3.5-1 Pressure build-up o r recovery tests
In producing oil fields, build-up tests are frequently used in place
of constant-flux pumping tests, the analyses o f which require uniform
initial head distribution in the reservoir (Matthews and Russell, 1967).
It is also a common practice in water well testing to record the head
recovery in the wellbore after a constant-flux pumping/injection period.
Analysis of the recovery data produces a check value for the
transmissivity (Lohman, 1972).
The recovery upon stopping production is mathematically expressed by
superposing the temporal variations in head as a result of hypothetical
injection and continuing production at the same point and rate (Theis,
1935). Assuming no change in the transmissivity and storativity at the
start of pumping and recovery, and utilizing the logarithmic approximation
(Equation 3-12),
where s' is the residual head (i .e. difference in head at the start of the
41
actual pumping period and any time during recovery), and t' is the time
elapsed since the pump was shut-off. The transmissivity value is
calculated from the slope of the data plot (s' vs. log t/t'). It should
be noted that the influence of well losses developed during the actual
pumping period propagates into the recovery period in the form of a delay
(in the actual process) during which formation pressure at the wellbore
face equilibrates with the pressure in the packer-isolated interval and
reaches the theoretical level predicted by the Theis equation (Equation
3.9).
3.5.2 Step-drawdm tests
The optimum well yield, the total drawdown at a desired pumping
rate, and changes in the efficiency of a well after being used or
developed are the essential information needed in the design of production
and drainage wells and well fields. The necessary information can be
extracted from a step-drawdown test, originally suggested by Jacob (1947)
to quantify the well loss. The standard procedure to conduct this test is
to record the total drawdown while increasing the pumping rate in stepwise
manner. The well loss constant (and the exponent) are calculated from the
slope of the data plotted in the following formats by rearranging: a)
Equation 3.17 as
and b) Equation 3.18 as
With the latter method the linear formation loss constant is repeatedly
estimated until a straight line of slope (n-I) is obtained.
The actual extent and pumping rate dependence of the head loss
domains of Equation 3.16 are not considered in the graphical solutions.
This is probably the main reason that tests in water wells yield values of
the exponent as high as n r 2 . 5 (Rorabough, 1953) and even upto n = 3 . 5
(Lennox, 1966). Theoretically the value of this exponent should equal 2
for exclusively linear flow as well as for linear flow with an abrupt
transition to fully turbulent nonlinear flow within the medium. The
existence of a long transitional nonlinear flow domain should reduce the
value to less than 2. Any changes between the test steps such as stress-
induced permeabi 1 i ty reduct ion around we1 ls, particularly in
unconsolidated media, may substantially contribute to the deviations cited
in the literature.
Well efficiency
Since the linear formation loss is an inevitable consequence of
fluid movement through porous media, we1 1 efficiency is referenced to this
loss as (Rorabough, 1953)
BQ Ew- - S
{ BP= s, I:; (3.23)
According to this definition, the efficiency of a well can be improved by
reducing the well loss and especially the nonlinear formation loss. This
43
can be achieved by applying methods such as gravel packing and improving
screen design in water wells (Kruseman and de Ridder, 1970), acidization,
sand propping and hydraulic fracturing in oil wells (Baker, 1955; Matthews
and Russell, 1967) and enlarging well radius in general.
3.5.3 Geotechnical permeability tests
The need for in-situ determination of permeability arises from the
presence of heterogeneities in rock masses that cannot be tested in the
laboratory at the scate/comp1exity with which they contribute to the bulk
permeability. From the geotechnical point of view, in-situ information is
crucial to delineate the distribution of seepage forces around engineering
structures (Cedergren, 1988). In this context, field permeability values
are obtained through we1 1 testing, general ly based on steady-state
approximation. This is adequate in engineering design, especially at
shallow depths of investigation where the well head stabilizes relatively
fast due to high permeability o f fractures (Maini et al. 1972).
Constant-flux pumping/injection permeability tests are usually
conducted i n packer-isolated intervals in order to profile vertical
conductivity and/or concentrate on the depths/features o f interest along
the fractured wells. The pumping tests require expensive large wellbores
and are limited to saturated zones. Therefore the injection tests (also
known as water pressure or lugeon tests), although prone to the clogging
effect for there may be impurities in the test water (Cedergren, l988),
are routinely applied in site investigations.
Because of cost concerns and the desire to obtain a representative
44
sample o f local permeabilities, much effort is focused on single-well
tests. The permeability derived from the single-well injection tests is a
local value (Louis and Maini, 1970), even for a single fracture, and only
the fracture(s) directly intersecting the wellbore influence the results
(Zeigler, 1976; Rissler, 1978). Considering the latter and assuming that
fractures occur in orthogonal sets, wellbores are recommended to be
drilled normal to each set in order to derive permeability independently
(Louis and Maini, 1970; Maini et al. 1972). The permeability values can
then be used either in a discrete (statistical) fracture network or in an
anisotropic continuum model to study seepage and the alternative remedies.
Problems and a e t h d of analysis
The flow rate response of a packer-isolated fracture under high
injection pressures may be significantly altered, in addition to well
losses, by (Louis and Maini, 1970; Maini et al. 1972; Zeigler, 1976;
Cedergren, 1988): a) the enlargement of fracture aperture; b ) leakage of
packers; and c) redistribution and/or washing out of filling materials.
Assuming the confining blocks and the fracture to be extensive, aperture
changes are limited to the elastic deformation of the matrix if injection
pressures are below the overburden pressure. Low pressures also help
control the leakage problem and the nonlinear formation loss. In general
practice the causes altering the flow rate response to pressure increment
are recognized from various nonlinear signatures on the flow rate-pressure
graphs drawn from multistage injection test data. The graphic format o f
Equation 3.21 i s more appropriate when used as for the step-drawdown
45
tests.
In the range where the nonlinearity is due solely to well losses,
the analysis of single-well permeability tests can be based on the
formulation of the head loss components (Equation 3-16). For a steady,
radial, two-regime flow (i.e. one in which linear and nonlinear domains
co-exist as specified for Equation 3.18) through a rough fracture, the
relationship between the net head change in the active well and flow rate
is approximated by (Rissler, 1978)
Similar forms of this equation were also derived by Rorabough (1953),
Baker (1955) and Bruin and Hudson (1955). The expressions for the
conductivity and friction factor in the linear and nonl inear formation
loss components (Equation 3. 24) , respectively, are selected from Table 2.1 according to the relative roughness of the fracture. Details of the
derivation in a fully explicit form of Equation 3.24 and the selection of
the roughness-dependent parameters can be found in Section 4.5 for the
generalized formulation o f flow through arbitrarily oriented fractures.
3.6 Low-perreability W i a tests
It is contemplated that hazardous wastes might be disposed in low-
permeability (crystalline, fractured) media at great depths. The high
risks associated with the containment of these wastes in the repositories
demand different well testing procedures and analysis methods for
hydraulic characterization of such unusual media. A thorough description
46
of the groundwater system (under probable field gradients) and therefore
of the effective fracture network is essential for an accurate portrayal
of the migration patterns for the released contaminants. Well testing in
this context is focused on defining the network mainly at the scale of a
constituent fracture (Wilson et al., 1979; Doe et al., 1982).
Transient single-well test methods (i.e. constant-head injection,
slug and pulse tests) are best suited for estimating the aperture, extent
and connectivity of the packer-isolated fracture(s) in low-permeability
media (Wilson et al., 1979; Doe and Remer, 1980; Doe and Osnes, 1985).
These parameters are combined with other geometrical observations such as
orientation and spacing to form statistically equivalent networks (Doe et
al., 1982).
3.6. Constant-head injection tests
These tests can be rapidly applied over a wide range of
permeabilities and are free o f wellbore storage effects (Doe and Remer,
1980). The test procedure involves injecting fluid into a packer-isolated
section under constant-head and recording the f l o w rate decline. Employing
the analogy between a single fracture and a confined aquifer (Doe et al.,
1982), the basic solution becomes the same as that of constant drawdown
tests in extensive confined aquifers (Jacob and Lohman, 1952)
Q(t) = ~ z T s G - (ST::) where G is the constant-head well function and s is the induced head
change in the active well. The definition and the numerical values o f the
47
function G and the associated type curve (log G vs. log ~ t / s r Z ) are
given by Lohman (1972).
A data plot (log p vs. log t) from a single fracture test may
reveal up to three temporal phases: a) an initial infinite response
period; b) a steep decline signalling a (partly) closed or low
permeability boundary; and c) stabilization indicating induced leakage or
network connection (Doe et al., 1982; Doe and Osnes, 1985). The
transmissivity and storativity of the fracture are obtained using curve
matching of the first section o f the data plot. The distance to the
(closed or constant-head) boundary can be determined from an empirical
relation such as that of Uraiet and Raghavan (1980)
where r, is the distance to the equivalent circular boundary and t, is
the time at which the boundary is felt.
The main limitation of this test methodology is the difficulty of
obtaining earlytime data in rigid and/or finite fractures (Doe and Osnes,
1985). Additionally, flow rate dependence of the well losses and time
dependence of fracture opening complicate data analysis from the early
response period.
3.6.2 Slug and pulse tests
Both tests consist of instantaneous elevation or lowering of the
head i n the test interval and monitoring the decay or recovery,
48
respectively. Slug tests are conducted by addition or removal of a known
volume into the test section through the open tubing whereas pulse tests
involve, pressurizing or de-pressurizing a packer-isolated interval. This
difference in producing the head change translates into different boundary
condition expressions. Accordingly the flow rate, for example, into the
fracture equals (Bredehoeft and Papadopulos, 1980): a) the rate of fluid
volume decrease in the open tubing during slug tests, i.e.
and b) the rate of volumetric expansion of fluid in the pressurized
section during pulse tests, i.e.
where r,: radius of the wellbore in the test interval,
r,: radius o f the open tubing,
v,: initial fluid volume in the test interval, and
H*: head in the tubing or test interval.
The analytical solution simulating the head transient during a slug
test in a finite diameter well in a confined aquifer is of the form
where H,' is the initial head in the tubing and F i s the pulsed-head well
49
function the definition and numerical values of which were first
introduced by Cooper et al. (1967). This solution is also valid for pulse
test response when, from Equations 3.27 and 3.28, the following
substitution is made
Successful applications o f this solution for both tests require limiting
the head increments to the lowest possible level to prevent fracture
opening and to minimize well losses during the early period.
The field observations can be evaluated by matching the data plot
(H'/H,' vs. log t ) to one o f the type curves (H'/H: vs. log ~ t / r : ) . Data
collected during the 50 to 80 % decay period is sufficient for this
purpose (Bredehoeft and Papadopulos, 1980). However, type curves for large
variations of the group parameter, s (Equation 3.29) are very
similar in shape and therefore transmissivity is the only reliable
estimate from these tests (Cooper et al., 1967).
The fracture volume influenced by these tests at the end of the full
decay of a given head increment is a function of the radius of the
tubing/wellbore and storativity o f the fracture. On the other hand the
length of time required for percent decay is solely determined by the
transmissivity for a given head increment, tubing/wellbore radius and
storativity. The range of transmissivity over which these tests apply is
determined by this length o f time which should be short to be practically
observed, but not too short to obtain sufficient data free of wellbore and
50
instrumental effects. In this sense Pulse tests are adequate for very
tight fractures (egg. 2b< 20pm) and are therefore complementary to
constant-head injection tests (Wilson et al., 1979).
The closed boundary i s felt as stabilization of head at an
incomplete decay whereas a constant-head boundary should accelerate full
decay. The distance to the equivalent circular boundary o f a single
fracture can be estimated from the total volume change at the time the
boundary is felt.
Pumping well-head I
Nonlinear . -
flow domain .
. . . . . . . .
. . . . .
Linear flow domain . . . . . . . . . . . .
. . . . . . . . . .
Figure 3.1. Radial half-section through the cone o f depression developed upon pumping from a confined aquifer/fracture under the Theisflhiem assumptions. Note the actual (A) and theoretical (T) head profiles, the components o f the total head loss, and the corresponding flow domains.
4-1 Introduction
The review in Chapter 3 demonstrates that the attitude as wet 1 as
hydraulic properties of the fractures intersecting an active well may
become important system parameters when the data being evaluated is from
the active we1 1 (and the early test period, if the response is transient).
The main purpose of this chapter is to develop analytical we1 1 test models
for evaluating such data. A brief account o f the most relevant literature
is given in Section 4.2. The methodology of the model development
described in Section 4.3 provides a firm ground on which the problem is
formulated. The analytical solutions to the general diffusion equation
developed in Section 4.3 are presented in Section 4.4 for transient and
steady constant-flux tests. At the end of Section 4.4, potential theory is
utilized to delineate the flow network and the zone of influence in acute
systems during tests under initially non-uniform head conditions. The
1 ikel ihood of nonlinear flow regime and its influence on the observed head
changes and hence the aperture (or transmissivity) predictions are
addressed in Section 4.5 where a general formulation is introduced.
Section 4.6 provides an overall discussion for various models proposed and
also includes a subsection addressing the natural extension of the basic
concepts to refine estimates of other well test models.
4.2 Previous studies relating to acute systems
No well-test models that consider the system intersection angle as
a variable that controls the intersection area and alters the flow pattern
53
and head distribution in the fracture were encountered during the course
o f this study. The models based on a horizontal or vertical fracture
intersecting a wellbore (such as outlined in Zeigler (1976) and Gringarten
(1982)) form the end members of a general system model in which both
fractures and wellbores can change attitude. Another end member was
conceived by Louis and Maini (1970) assuming the wellbore to be orthogonal
to an inclined fracture.
Cinco-Ley (1974) developed a unique model simulating the transient
response to a constant-rate production of a slab reservoir as a function
of the fracture-we1 lbore intersection angle. Since the fracture in this
model was considered to be a plane sink, the intersection angle merely
changed the reservoir volume to be drained. Through a numerical simulation
of the steady injection tests in unsaturated zones, Rissler (1978)
evaluated the influence of fracture inclination in acute systems composed
of a vertical wellbore intersected by an inclined fracture as well as in
orthogonal systems with inclined fractures. In order to account for the
dominance of the intersecting fractures on the near wellbore flow,
Karasaki (1987) conceptualized the fractured medium as two concentric
zones: the inner zone composed of a vertical, horizontal or inclined
intersecting fracture (or fractures with identical hydraulic properties
and attitude) and the outer zone of a three-dimensional, well-
interconnected fracture network. For the case o f an inclined fracture, the
inclination was assumed to produce a three-dimensional connectivity
between the two zones leading to a spherical flow field in the outside
zone reducing to radial flow in the inner zone (or vice versa).
Theoret ica7 basis
The Darcy law (Equation 2.2) in vectorial form can be written as
where u' is a unit vector. The velocity and gradient vectors at any point
are co-linear in isotropic media (i.e. where conductivity is a scalar).
With regard to their definition, the gradient vectors are normal to the
equipotential surfaces (del ineated by h =const. ) and the stream1 i nes are
tangent to the velocity vectors. Therefore, all equipotential surfaces and
(instantaneous) stream1 ines intersect each other at right angles (Rouse,
1961). This deduction also holds for turbulent two-dimensional flow in
homogeneous, isotropic media (Bear, 1972). Furthermore, in such media, the
streamline pattern in an induced flow field is solely controlled by the
geometry of the flow domain for an initial head distribution. These
theoretical considerations underlie the assumptions necessary to validate
specific flow patterns conceptualized to form during well tests.
The inner boundary of a convergent/divergent flow field in a
fracture is the intake/outlet section at the fracture-wellbore
intersection. This section coincides with the innermost equipotential
surface to which streamlines will be normal. Provided that the fracture
void space is homogeneous, isotropic and extensive and that the head
distribution before f l o w is induced is uniform, streamlines will remain
normal and hence straight. Since the induced head distribution along all
55
these streamlines will be identical, equipotential surfaces pass through
points of equal distance from the wellbore face. Again for this reason,
most analytical models of well tests are formulated in plane-polar
coordinates which is best suited to describe the head distribution along
straight radial streamlines.
Description of the conceptual muiel
It is clear from the preceding discussion that the geometry of the
inner boundary determines the streamline pattern and subsequently the head
distribution. The boundary geometry is, in turn, determined by the angle
o f intersection between a fracture and the wellbore. Here, the fracture is
thought of as a parallel plate conduit, an idealization theoretical
aspects of which are extensively discussed in Chapter 2. For parallel and
orthogonal intersections (Figure 4.l.a), the inner boundary outline are of
linear and circular forms, and the resulting streamlines are parallel and
radial, respectively (Figure 4.l.b). For acute intersections, models based
on assumptions constraining streaml ines to these forms provide approximate
estimates, the accuracy o f which could not be quantified previously.
Acute fracture-wellbore intersections (Figure 4.l.a) produce
intake/outlet sections that are elliptical in plan view (Figure 4. Lb).
For a given wellbore radius, as the ratio of the axes of the ellipse
varies with the intersection angle, the streamline pattern is not of a
fixed form but a general one (Figure 4.1 .b) . The streaml ine pattern, given that the head distribution is initially uniform, is independent of the
inclination of the fracture-wellbore system with respect to datum.
56
Just i f icat ion f o r single fracture system mdel s
Fluid exchange between fractured rock masses and wellbores generally
takes place via a limited number of fractures (Sharp and Haini, 1972).
These fractures contribute to the total flow rate in varying proportions
and form fracture-wellbore subsystems. The contribution of each subsystem
is determined by the attitude and hydraulic properties of its fracture
component. For example, the fracture with the largest aperture in the test
section dominates the flow rate (Rissler, 1978; Doe and Remer, 1980) given
that all the fractures are parallel and identically connected to the flow
network.
In well testing practice, it is equally essential to understand the
interactions at the subsystem level. For example, oil wells are often
terminated at the interval where the first significant mud loss occurs
generally because of the presence of a single dominant fracture (Baker,
1955). Hydraulic fracturing for the stimulation of fluid recovery wells
typically creates a single fracture dominated f l o w condition. Another
common case is where a single fracture is packer-isolated in the test
section.
Significance o f to tal head i n comergent/divergent flows
For viscous, incompressible, established flows through uniform
conduits, the (weighted-mean) total head across a flow section is given by
the modified Bernoulli equation (Rouse, 1961)
57
where H: total head (or total mechanical energy per unit weight),
F2/2g: velocity (or kinetic) head,
a: kinetic energy correction factor for non-uniform velocity
profiles, and - v: average velocity calculated from the continuity condition.
In such conduits, the difference in the total head between two sections i s
AH=Ah (4-3)
which verifies that the driving force of the fluid movement, i.e. the
gradient, can be conveniently expressed in terms of the hydraulic head
rather than the total head. However, when the area of flow at each o f
these sections is different, both the magnitude and the profile of the
velocity varies,
In convergent/divergent flows, velocity variations occur regardless
of the conduit geometry. For example, fluid particles flowing toward a
well through a uniform parallel plate accelerates in response to narrowing
flow area to maintain the continuity in volumetric flow rate. This
phenomenon is accomodated by the continuous transformation between
hydraulic and velocity heads. Subsequently, the irrecoverable losses in
the energy of fluid can be directly depicted only in terms of the total
head (Figure 4.l.c). In conclusion, the Darcy law (Equation 2.2 or 4.1)
and other flow equations discussed in Chapter 2 must be re-written in
terms of the total head in order to be adapted to the formulation of well
58
flow problems, especially wherever the velocity head differences are
expected to be significant.
4.3 Uathematical formlation o f the problem
The first step i n the development of analytical models to simulate
well tests (and any subsurface flow problem) is to write the diffusion
equation (e.g. Equation 3.5) in appropriate coordinates. The purpose of
this section is therefore to introduce the derivation of this fundamental
differential equation, for the present problem, for which analytical
solutions will be sought.
In a (convergent) flow field displaying the general streamline
pattern, the net accumulation of fluid mass during a finite period within
an elemental volume bounded by (concentric) equipotent ial surfaces may be
expressed by
where I' is the perimeter o f the equipotential surfaces passing through
points at distances 1 and 1+A1 from the wellbore face, and 6 i s the
average velocity normal to these surfaces. Obviously, it is not possible
to proceed from Equation 4.5 unless the perimeter of the equipotential
surfaces can be formulated as a function of the distance from the we7 lbore
face. The derivation of the perimeter expression for the general
streamline pattern is presented in Appendix A. The resulting functional
relationship is (Equation A.15)
rll=z=
where e is the perimeter of the ellipse divided by ir . Substituting Equation 4.6 in Equation 4.5 and dividing both sides by the elemental
volume, I'll (2b) Al,
Passing to the differentials (and remembering tha t F'[J>Flll+4J,
Neglecting the density variations along streamlines (i .e. 0 ) and
substituting the velocity term with the Darcy law (Equation 2.2) and the
right hand side with Equation 3.2 (both re-written i n terms of the total
head)
which is the desired diffusion equation. By taking dh/dt=aH/at, it is
also assumed that the velocity field is steady. Since, by the definition,
the velocity term in Equation 4.2 represents an average of the f l o w
section, Equation 4.9 can also be written as
60
which is the counterpart of Equation 3.7. However it should be noted that
the solution domain in Equation 4.10 starts from the wellbore face whereas
in Equation 3.7 from the wellbore axis.
4.4 Analytical models o f constant-flux tests
In this section, solutions of Equation 4.10 corresponding to the
Theis and Thiern equations (Equations 3.9 and 3.15) are presented. The
assumptions underlying these solutions are specified in the following
(using the same order as in Section 3.3):
a) the fracture is homogenous and isotropic in aperture and
roughness (i-e. parallel plate idealization is valid), of infinite
extent (i.e. boundary effects are not felt during testing), and o f
arbitrary inclination,
b) the matrix of the confining blocks behave as impermeable (under
induced vertical pressure differentials),
c) the well intersects the fracture at an arbitrary angle,
d) the hydraulic head distribution prior to testing is uniform,
e) the streamline pattern is not axismetric but of a general type
(as conditioned by a-d), and
if the head response is transient,
f) water is instantaneously released frowenters into the storage as
the hydraulic head descends/rises, respectively, and
g) the test section is a short, packer-isolated interval so that the
wellbore storage is negligible, but the wellbore has a finite
diameter.
61
Attention needs to be drawn particularly to the assumption (e) which
provides originality to the following solutions. As the equipotential
surfaces approximate circular rings (i.e. e+l=l) at some distance from
the wellbore face (Figure 4.l.b), it may be allowed to assume more
reasonably that the fracture i s of finite extent and connected to a
laterally extensive fracture flow network (Karasaki, 1987). Naturally, any
differences between the hydraulic properties of the network and the
fracture should be reflected in the observed response pattern.
4.4-1 Transient head
The preceding assumptions lead to the initial and boundary
conditions,
where the initial head level is chosen as datum (i-e. h , = O ) . Employing
the Laplace transform method and the initial condition, the diffusion
equation (Equation 4.10) can be reduced to the (zero order) modified
Bessel equation
62
where K(1 , Z) =g [ ~ ( l , t) ] and F is the transform variable. The general
solution of Equation 4.12 is (Tuma, 1987)
K = ~ I , [ k ( e + l ) I +44 [ R e + l ) I (4.13)
where I, and rc, are the (zero order) modified Bessel functions, and A,
and A, are constants. From the first boundary condition,n(=, 3 =hO/F=o
and since %(-I = O and I~(-) =', 4 = 0 . Hence
i7=A.Jo [ E ( e + l ) ] (4.14)
Applying the second boundary condition and taking the derivative of
Equation 4.14 with respect to 1
where K, is the (first order) modified Bessel function. Substituting A,
(Equation 4.15) and (defined i n Equation 4.12) in Equation 4.14
Employing the inverse transform o f the function in the main brackets given
by Carslaw and Jaeger (1959) results in the desired solution
where
and Z is the acute intersection we11 function, x is the integration
variable, J,, J,, Y, and Y, are the Bessel functions o f zero and first
order. A solution of the same form as Equation 4.17 was previously applied
to the radial flow in orthogonal wellbore-aquifer systems with finite
diameter wells by Hantush (1964). A short table o f values for a function
that equals (1/4x) Z was presented by Ingersol 1 et a1 . (1950). Thus, the transmissivityof a fracture forming an acute intersection with the active
well can be predicted by matching the well head (or drawdown) vs. time
graph to the type curve (Figure 4.2) , drawn from these values
corresponding to r / o = ~ (i.e. the wellbore face). On Figure 4.2, the
acute intersection type curve i s compared with that of the l ine source
solution of Theis (1935) in order to demonstrate another advantage o f the
present solution.
4.4.2 Steady head
The diffusion equation (Equation 4.10) reduces to the Laplace
equation when it can be assumed that i W / a t = o
The boundary conditions for steady constant-flux tests in acute systems
are
H ( L ) =HL
where L is the distance to the point of measurement. Following the
standard integration procedure for these conditions (Equation 4-20),
The zone o f inf7uence i n unsaturated zones
In many cases, an acute system is created by a vertical wellbore
cutting through an inclined fracture. When the initial heads are not
uniformly distributed, the pattern of streamlines will deviate from that
of the uniform heads (Figure 4.1.b). This situation is frequently
encountered during injection tests for foundation permeability
determination in unsaturated zones. It is therefore of interest to
delineate the modified network of streamlines and equipotentials to assess
the compounded (asymmetric) bias i n the fracture surface coverage in such
test settings.
The influence o f fracture inclination can be simulated by the
(fictitious) gravity flow (Louis and Maini, 1970). Utilizing the linearity
65
o f the Laplace equation (Equation 4-19), the resultant head distribution
is obtained from the superposition o f the solutions for the source
(Equation 4.21) and gravity flows
where ~ ( 1 , 8) and 9 are as def f ned in Appendix A (Equation A.8 and Figure
A.1, respectively), and h is the inclination of the fracture from the
horizontal. Similarly, the stream function for the superposed source and
gravity flows is
where
tan0 f' E ( x , a ) +Itan-I(-) rcl,e) =- sins sins
~ ( 1 ~ 8 ) , r and a can be obtained from Equations A.9, A.16 and A17,
respectively, and E ( ~ , c r ) is the incomplete elliptical integral of the
second kind. As the point of stagnation exists at 0=0, the general
equation defining the zone o f influence in acute systems is found t o be
+(l,e) = O (4.25)
An example of the zone of influence upon injection in acute and orthogonal
systems is illustrated in Figure 4.3.
4.5 Total drawdown i n steady-state, two-regiae flows
The solutions (Equations 4.21 and 4.17) developed in Section 4.4 are
66
based on the assumption o f laminar flow (i.e. validity of Darcy law).
Therefore the drawdown predicted by these solutions represents the linear
formation loss component of the total drawdown (Equation 3.16).
Considering that the turbulent flow domain is generally limited to near
wellbore region where the velocity is largest and that the onset of
turbulence is facilitated during injection by the intersection geometry
(Haini et al., 1972), it is desirable to pursue equivalent solutions for
the turbulent flow conditions. This is particularly important for steady-
state injection tests where focus is on the near wellbore environment.
The relationship between the total head gradient and the average
velocity in a fully turbulent flow domain is given by the Missbach
equation (Equation 2.23)
where 11 is as defined in Equation 2.27. Substituting this in the equation
of continuity (Vennard and Street, 1982)
Q = ~ A { A = 2 x ( e + l ) (2b) (4.27)
and integrating between any two poin ts within the domain yields
where S, is the nonlinear formation loss component of the total drawdown
(Equation 3.16).
With analogy to the pipe flow studies (Vennard and Street, 1982),
exit and wellbore losses in fracture-wellbore systems can be modeled as
67
linear functions of the velocity heads at the inlet/outlet section and
within the wellbore, respectively, as in Equation 3.24 (Rissler, 1978).
Hence, introducing the concept of the critical distance and writing the
conductivity in terms of the aperture, the expression of total drawdown
for the entire flow system results
where f: conductivity modification factor (Equation 2-20),
1,: distance from the wellbore face to the outer boundary,
1,: critical distance at which an abrupt transition between
laminar and fully turbulent flow conditions is assumed to take
place, and
f, and e,: empirical exit/entry and wellbore loss coefficients,
respectively.
The critical distance can be estimated from
where Re,, the critical Reynolds number, can be computed as explained in
Section 2.4.2 for a given estimate o f the relative roughness of the
fracture. This estimate is also necessary to compute the friction factor
68
in the nonlinear flow domain (Table 2.1) and the conductivity modification
factor in the linear domain (Equation 2.20). Therefore, the predictive
ability of this semi-analytical model of steady, constant-flux tests
(Equation 4-29), within the limitations of its assumptions, is primarily
based on the estimate of the relative roughness.
The variations in the magnitude of the exit/entry loss coefficient
(Equation 4.29) as a function o f the intersection angle is investigated by
the laboratory study described in the following Chapters 5 and 6. In the
meantime, differences in aperture predictions at the same injection head
for various intersection angles are presented in Figure 4.4 by assuming
that the total head at the wellbore face is known.
4.6 Discussion
A conceptual study undertaken to understand how the intersection
angle between a fracture and a we1 lbore modifies the response lead to both
transient (Equation 4.17) and steady-state (Equation 4.21) solutions for
constant-flux tests in acute systems. No further assumptions other than
those of the Theisflhiein equations (Equations 3.9 and 15) were introduced,
whereas the vertical we1 lbore and horizontal aqu ifer/fracture assumptions
were lifted. The test models can simulate the head response in the entire
range of intersection angles. These models offer a better physical
understanding of flow around the fractured wells and exemplify flow
analysis where the streamline pattern is asymmetric with respect to the
origin. However, the use of the acute intersection models are as easy as
their radial equivalents. Various aspects and practical implications of
69
these models are discussed below.
The type curve depicted in Figure 4.2 indicates that, for all
intersection angles, the transient response pattern is identical in the
domain of the dimensionless time, t / 02 . It i s also shown on Figure 4.2
that for t/02z25, the acute system response coincides with that
predicted from the Theis (line source) solution (Equation 3.9). However,
the real time t at which this is realized are delayed with reference to
the orthogonal system by a factor of (es/%o)2 as the intersection angle
decreases. In other words, the early period where the intersection angle
dominates the response is longer for smaller angles. The delay factor
( ~ ~ / e , , ) ~ reaches an order of magnitude at P - 1 2 O .
The solution for steady response (Equation 4.21) predicts the linear
formation loss, and therefore is directly incorporated in the model of
total drawdown (Equation 4.29). The streamline-equipotential network that
develops during tests under steady, laminar flow conditions in unsaturated
zones can be analytically simulated using the formulation presented in
Equations 4.22 to 4.25. Interestingly, the zone of influence is
practically unchanged with the intersection angle (Figure 4.3). This
observation is important in that individual permeability test results from
acute systems with different intersection angles can be directly
integrated in a rock mass model.
The development o f a model (Equation 4.29) to evaluate the steady,
constant-flux tests in acute systems with rough fractures is of utmost
practical significance. Based on this model, Figure 4.4 illustrates that
70
the fracture aperture for an acute system is overestimated by the
orthogonal system model (Equation 3.24). The magnitude of the error at any
head level is constant for a given intersection angle, and varies only
slightly for different combinations of test variables because of the
remarkable sensitivity of the total head loss to the aperture. Obviously,
this magnitude will be somewhat different if one references the
predictions to the head inside the wellbore and includes the intake/exit
loss for that intersection angle.
Looking at Figure 4.4 from a cost efficiency perspective, it becomes
very clear that well losses can be significantly prevented by reducing the
intersection angle using oriented drilling at the production levels.
Similarly, efficiency of wells drilled by the identical procedures into
the sane medium may differ substantially if the fractures dominating the
flow rate form different intersection angles in each wellbore.
Furthermore, the necessity o f multiple orthogonal drilling in
determination of anisotropic permeability is unjustified as a single
wellbore may be sufficient for this purpose.
Equivalent radius concept i n the evaluation of single-well tests
Beside constant-flux tests, there are a variety of other single-well
tests (Chapter 3) evaluation of which equally needs consideration of the
fracture-wellbore intersection angle. Although specific models for each
test type may be developed, at least through numerical inversion of the
solution in Laplace space (Stehfest, 1970), this is beyond the scope of
the present study. However, it is noteworthy that the equivalent radius
71
concept as defined in Equation A.15 may be used to refine the estimates of
the available solutions of the single-well tests. According to this, an
acute fracture-wellbore system i s replaced with an orthogonal system which
has proportionally larger wellbore radius. Thus the influence o f the acute
intersection is reduced to a simpler problem of changing the wellbore
radius. This approach can be extended to other solutions based on more
complicated conceptual systems such as the composite model (Karasaki,
1987) and the equivalent homogeneous medium model with wellbore storage
and skin effects (Gringarten, 1982).
Figure 4.l.a) Parallel, orthogonal and acute fracture-wellbore intersections along a vertical wellbore; b) corresponding intersection out l ines (linear, circular, elliptical) and streamline patterns (parallel, radial, general) ; and c) sketches of total head prof i les upon inject ion under initially uniform head and laminar f l o w conditions.
Q : 0.5*10-' mJ/s
r,: 0.027 rn
2b: 0.540-' rn
v : I.O*IO-' m2/s
Figure 4 . 3 . The zone of influence upon injection i n t o acute and orthogonal systems P
with inclined fractures in unsaturated zones.
0 m o o -
5 LABORATORY STUDY OF FLOW THROUGH ACUTE FRACTURE-YELLBORE SYSTEUS
5.1 Introduction
The total head loss that occurs during single-well, constant-flux
injection/pumping tests conducted through acute systems is formulated in
Section 4.5 (Equation 4.29). As a semi-analytical equation, this:
a) provides a basis for a controlled laboratory design and
systematic analysis intended to develop a better understanding of
flow mechanics in acute systems; and
b) necessitates comprehensive testing to establish a fully
functional expression.
Accordingly, the objectives of the laboratory study are defined:
a) to test the validity of the conceptual streamline pattern and
study the f l o w interactions at the intersection under both pumping
and injection conditions; and
b) to determine the empirical relationship between the intersection
angle and exit/entry losses related to sudden changes in flow area
and direction at the intersection.
PrevSous experimental work and the theoretical background are
reviewed in Sections 5.2 and 5.3, respectively. Then the design features
and fabrication procedure o f the acute system models and the box frame are
explained usfng detailed drawings (Section 5.4). Description of the
experimental set-up also includes water circulation and instrumentation
components. Finally, experimental design and test results are outlined
(Section 5.5). Analysis and interpretation of experimental data are
presented in Chapter 6.
77
5.2 Physical models of orthogonal systems
Experimental research on flow through fracture-wellbore systems is
focused on identifying causes of head losses in order to improve
predictive radial flow models and/or minimize the well losses, for a wide
variety of industrial applications. The influence of turbulence and
surface roughness in radial flow was first studied experimentally by Baker
(1955). He devised a physical model made of concrete to specifically
simulate fractures in limestone reservoirs. In his mathematical derivation
of a two-regime radial flow expression, parallel plate conductivity was
employed in the linear regime, and roughness was expressed as a
coefficient in the turbulent regime. This coefficient and the critical
Reynolds number were determined as empirical constants.
In an attempt to improve water pressure (lugeon) test analysis,
Rissler (1978) tested the validity of theoretical and empirical one-
dimensional f l o w expressions (reviewed in Chapter 2) in describing
divergent, two-regime, radial flow. His physical model consisted of a
rigid, open and rough (radially isotropic) fracture with apertures as
small as 10" m. A satisfactory agreement between measured and calculated
pressures was reported. Atkinson (1987) extended t h i s to convergent radial
flow in rough deformable fractures and addressed the problem of mine
dewatering using vertical drainage wells.
An extensive experimental study was undertaken by Murphy (1979)
mainly to explore the influence of acceleration on convergent laminar and
turbulent flow predictions. Tests with laminar flow through a rigid,
smooth, open fracture verified pressure profiles obtained from a numerical
78
solution of Navier-Stokes equation (Equation 2.1) in radial coordinates.
The study by Murphy (1979) is specifically concerned with the flow
mechanism near the outlet of a geothermal recovery well.
Experimental set-ups used in these earlier studies have been
reviewed in detai 1 to provide a viable, versatile and functional design in
the current study.
5.3 Theoretical basis for model design
Model design is centred around the Reynolds number concept (Equation
2.19) which:
a) allows generalization of the results obtained from experiments
with physical models designed for one set of variables, and from a
limited number of test runs;
b) helps set an optimum range for flow rate and fluid temperature
that will produce Reynolds numbers typical o f field experiments; and
c) provides flexibility in determining model dimensions (effective
flow length, well radius, aperture) and boundary pressures, that
will produce the targeted Reynolds numbers.
Simulating the flow processes using the Reynolds number is reliable under
hydraulically smooth conditions for which the friction factor is
approximated as a function of only the Reynolds number for both (laminar
and turbulent) flow regimes (Table 2.1). Otherwise, the friction factor,
and therefore associated head losses, are partly or totally independent of
the Reynolds number. The model testing should therefore be conducted under
hydraulically smooth conditions (Table 2.1) unless the model fracture is
morphologically smooth (i .e. k / ~ , = o ) .
5.4 Description o f experimental set-up
The overall experimental set-up (Figure 5.1) used in this work
consists of a) three fracture-wellbore system models separately fastened
to b) a steel box frame, c) scheme for water circulation, and d)
instrumental components. It should be clear from the experimental
objectives that the set-up is expected to simulate and control the
conditions on which the mathematical model (Equation 4.29) is based. These
can be stated as:
a) the fracture is rigid (i.e. insensitive to changes in fluid
pressure), of infinite areal extent, isotropic and homogeneous in
hydraulic conductivity;
b) hydraulic heads are temporally constant at the boundaries and
initially uniform over the confined flow domain; and
c) flow is isothermal.
The fracture properties are intended to produce straight streamlines, and
head distribution and flow temperatures are prescribed to ensure steady-
state tests. The following section explains how the above parameters are
satisfied and other design features are decided.
5.4.1 Design o f acute syste~ models
The intake/outlet flow area at fracture-wellbore intersections is
the key factor causing differences in flow responses of parallel,
orthogonal and acute fracture-wellbore systems (Figure 4.1.a). A plot of
80
normalized flow area vs. intersection angle (Figure 5.2) indicates that
this difference should become significant for angles less than 40.. To
capture these more pronounced effects of the intersection geometry, the
design and fabrication of two models with 10' and 20' intersection angles
were carried out for the laboratory study. An orthogonal model was used
mainly to verify the performance of the experimental set-up as a whole.
Figure 5.3 illustrates various design features of these models as
discussed below.
The areal dimensions of the models were determined on the basis o f :
a) an optimum effective length along which flow would fully establish and
significant pressure differentials would develop; and b) a wellbore
diameter which enables high Reynolds numbers to be reached at low flow
rates and boundary pressures. A flow length of 0.27 m (0.28 m for 10'
model) and a wellbore diameter of 25 m were found to be satisfactory.
Similar dimensions have been used in earlier models (Rissler, 1978;
Murphy, 1979), but in different combinations. The short flow length of the
models, however, requires the outer boundary to conform to the propagation
front of the straight streamline flow i n order to satisfy the assumption
of infinite fracture extent.
The acute angle between the we1 lbore axis and fracture plane, and
the obliquity of the intake/outlet f l o w section produce directional
variations in the streamline bending angle (i.e. in inertial loss) and in
the geometry of the section (i.e. i n flow resistance), respectively.
Coupled with contraction/enlargement of the section during flow exchange
between a wellbore and a fracture, directional dependence of injection
81
pressure profiles are likely very influential in preventing a straight
streamline flow pattern. This demanded a detailed study o f pressure
profiles along typical arrays (Figure 5.3.a), coinciding with expected
stream1 ines, at test Reynolds numbers, i .e. in addition to low rate tracer
injection tests which confirmed streamline flow visually.
Pressure holes were located so as to capture logarithmic variation
in pressure. A few holes in each model were located on symnetrical
streamlines at the same flow lengths in order to check aperture
uniformity. Wellbore pressures were measured at the well bottom parallel
to the wellbore axis and at two upstream positions (for 10' and 20'
models) to monitor flow structure and wellbore losses in inclined holes.
Such pressure holes would likely introduce a slight disturbance in
flow: an array o f pressure holes along a streamline will record an
increased disturbance downstream. In this study any such effect when
recorded fell within the uncertainty range o f pressure measurements, since
readings on two radial arrays (Figure 5.3.a-90') were almost identical
with those holes at the same radial distance.
Transparent acrylic (Plexiglass) sheeting was chosen as the model
material on the basis o f workability, flow visibility and cost. It was
decided to use a sheet thickness of 25 rm to help maintain flexural
rigidity and also provide adequate depth to support the pressure holes and
adapters (Figure 5.3.c-Inset). This also allowed for two central 0.22 m
diameter disks to be mounted in the same main plates in the design of the
20' and 90' models (Figure 5.3.c). This resulted in a more tedious model
fabrication but enabled machine drilling of the 20' inclined wellbore and
82
reduced material cost and labour. The long span of the inclined drill
section of the 10' system required the fabrication of a separate model
(Figure 5.3.b).
The intake/outlet section becomes off-set for any aperture, 2b,
other than that for which the well is drilled through the fracture.
Quantitative analysis of test results from each model is therefore limited
to that fabrication aperture, 2b. An aperture of 1 ma was decided upon as
a compromise between possible flow rates, pressure heads and desired
Reynolds numbers. The chosen aperture value is also within a range
encountered in nature (Bianchi and Snow, 1968; Chernyshev and Dearman,
1991).
If any morphological roughness is to be designed into the model
fracture, it should be isotropic and homogeneous, but these are
technically difficult features to provide. A relative roughness of 0.01,
for example, would limit tests to Re<5000 (Figure 2.2) i f theoretical
validity is to be maintained. The fracture surfaces were therefore left
smooth ( ~ / D , = o ) , in the models used here.
5.4.2 Steel box frame
Fluid pressure distribution and flow rate are extremely sensitive to
aperture changes near the wellbore where pressure differentials are
largest. Adjusting and maintaining aperture uniformity is therefore the
most vital requirement for the reliability o f the test results. A steel
box frame (Figure 5.4.a) was designed to act as an internally rigid system
under calculated test pressures. The resultant assembly, when the
83
plexiglass models were installed in the frame, simulated a fracture in a
rigid rock mass cut through by a wellbore.
The box frame contains two identical halves each consisting of four
tie-beams with high flexural rigidity welded in parallel to two H-beams,
thereby providing fixed ends. Each fracture model plate was fastened to
the frame with twelve steel shoes (Figure 5.4.a-Inset). These were
distributed evenly and symmetrically and not to block pressure holes in
any of the three models. The shoes allowed for alignment of boundaries and
uniform adjustment of the aperture. The frame design permitted the maximum
closure or opening of the fracture aperture to be calculated from measured
linear strains. A t maximum differential test pressures the frame allowed
an aperture change of only 0.01 m.
5.4.3 Fabrication procedure
The trial and error process of fabrication was eliminated by
computer aided design of the models and box frame. Most of the fabrication
was done by the Engineering Technical Services facilities of Memorial
University. Similar procedures were followed during fabrication of all
physical models. To extend the wellbore length and to accommodate the
push-in PVC connection pipe, thick plexiglass pieces were first fused onto
the external surfaces of the plates. To avoid off-centring of the drill
hole and to obtain the desired aperture, the model plates were clamped
together with the aperture laminae placed in between.
Whereas the 20* and 90' holes were machine produced, the 10' hole
was manually drilled through a guide hole. In this model, a 400 mn
84
inclined section was completed at the prescribed 10' angle by successively
enlarging a smaller guide hole. Some boundary irregularities at the
intersection resulting from drill vibration were smoothed using plexiglass
flakes dissolved in methyl chloride. No such problem was encountered
during fabrication of other models.
The large plates were cut to the theoretical outer boundary geometry
by a computerized lathe at the NRC Institute of Marine Dynamics. The
pressure holes were drilled 1.5 nm in diameter with square edges at
calculated coordinates for accurate measurements (Goldstein, 1983). The
holes were widened and threaded halfway for the pressure adapters (Figure
5.3.c-Inset). Finally the threaded holes for bolting the frame shoes were
drilled on external surfaces of the plates.
5.4.4 Water circulation system
The components of the water circulation system were designed such
that steady-state, pumping and inject ion tests could be conducted with
minimal change in the set-up (Figure 5.1). A polyethylene tank of 0.90 m
diameter was used to accomnodate the box frame and installed model. This
tank has side-wall access for inclined wellbore sections, drainage and a
thermocouple probe. The box frame, and hence the model fracture, was laid
horizontally in the tank to provide mechanical stability, and minimize
space and data analysis complications. The results, however, are
applicable to systems of any orientation with uniform and constant
boundary hydraulic heads.
Reduction of injection pressure below vapour pressure at test
85
temperatures seems inevitable unless downstream pressures are high enough
to compensate for the largest pressure drop at the entry vena-contracta.
Wellbore pressure under vacuum pumping is also below the vaporization
limit. This problem might be avoided by using a model tank that can be
sealed and pressurized. However the results of earlier experiments with
similar inject ion (Rissler, 1978) and pumping (Murphy, 1979) set-ups
suggested that such an expensive option was unjustified. In both injection
and pumping tests the vapour pressure limit was exceeded but no bubble
formation was observed even at the highest flow rates. Large amounts of
bubble formation would cause expansion and higher flow rate readings in
the pumping set-up, and would violate the applicability of the Bernoulli
equation based on the assumption of incompressibility (Equation 4.2).
In steady injection experiments, a net maximum head of 2.2 m was
reached by elevating the water tank. The required Reynolds numbers are
easily produced by this natural hydraulic head. Water suppl ied to the tank
was de-aired through air vents and stabilized by the constant head
apparatus. A valve located about 20 pipe diameters downstream from the
flow sensor was used to regulate the flow rate. Water level in the model
tank was kept below the level o f the fracture to produce constant and
uniform atmospheric pressure at the outer boundary.
For steady pumping experiments the model tank was allowed to
overflow to ensure constant water table level. A centrifugal (1/6 hp) pump
was sufficient to withdraw water at the same flow rates as used in the
injection tests. In both injection and pumping set-ups the upstream length
o f the pipe from the flow sensor was about 40 pipe diameters to a1 low flow
86
to be fully established. The wellbore itself was kept straight and
consistent in section for more than 20 diameters to eliminate flow
disturbances.
5.4-5 Instrumentation
Details of instrumentation and wiring are illustrated on Figure 5.1
in an overall experimental setting. The data acquisition unit consisted of
a plug-in card interfacing with a personal computer through a customized
software that acquired and stored data in the required format. All
measuring devices were connected to the card by an external terminal
panel. Fluid temperatures were measured by a subminiature thermocouple
probe inserted into the water in the model tank through its side wall. The
open circulation o f cold tap water provided very stable temperatures with
less than * 0.25 'C variation during a test run.
Five solid state piezoresistive pressure transducers were used for
both gauge and vacuum pressure measurements at twenty locations. Each
transducer was connected to four manometer tubes with a 5-way valve.
Injection/withdrawal flow rates were measured by a paddlewheel flow sensor
connected to a signal conditioner for computer interfacing. The pipe
assembly housing the flow sensor was made portable for easy rearrangement
between the injection and pump set-ups and for fast mounting alignment.
A full bridge circuit strain gauge measurement technique was used to
monitor any deformation of the fracture that may have taken place. Two
active gauges were fixed on central tie-beams to measure linear strain due
to their flexure and two other "duminy" gauges on an H-beam for temperature
87
compensation. All were covered with the protective coatings for underwater
operation. Strain gauge responses were calibrated by simultaneous direct
LVDT measurements of deflection due to step loading at the mid-span.
5.5 Experimental design and test results
The main parts of the experimental work were injection and pumping
tests of the model fractures at the fabrication aperture. In order to
detect Reynolds number dependent variations, each test was conducted at
several flow rates. A series o f test runs were performed at arbitrarily
chosen, and occasionally repeated, flow rates to prevent the development
of any systematic error in the readings. Water temperatures during each
series of runs was kept constant to allow direct correlations of pressure
measurements.
Nonlinearities or unexpected changes in observations during step-
drawdown and other multi-rate tests often result from a combined effect o f
hydraulic opening/closure and turbulence. The influence o f opening/closure
of the aperture can be evaluated by adjusting the aperture manually and
comparing the results with those obtained for the fabrication aperture.
Since the wellbore axis becomes offset in acute systems with any change in
aperture, it is also important to determine sensitivity o f flow mainly to
aperture change and somewhat to geometric structure at the intersection.
Therefore this has been made an integral part of the systematic testing
programme.
Each model was mechanically ad josted to the fabrication aperture and
the frame-model assembly then lowered into the model tank. Both injection
88
and pumping tes t runs were completed under i den t i ca l sett ings. PARKER
PRESTOLOK-type connecters (Figure S.3.c-Inset) with polyethylene manometer
tubing were ideal f o r f a s t mounting o f t h e models, de-air ing o f the tubes
and accessing background pressures. The model, wh i le i n the model tank
(Figure 5.1), was then re-adjusted t o the spec i f i c aperture value t o be
tested and the tes t ing procedure was repeated.
A t o t a l o f twelve t e s t series, each wi th an average o f s i x runs were
completed. The variables recorded dur ing each run and the parameters
character iz ing each ser ies are presented on Table 5.1 i n tes t ing sequence
and i n a format reduced t o consistent met r ic uni ts. Analysis o f the data
i s the subject o f the fo l l ow ing chapter.
[(r,,,, 2b. B), fmcture length, plananiy]
A: lnfake/outlet area r; Well radius
2b: ParalIeI plate aperture B: Intersection angle
'10 50 Intersection angle
Figure 5 . 2 . Rate of change o f intake/outlet area as a function of the intersection angle. Figures are normalized with re spec t t o the area of orthogonal intersections
with equal diameter wellbores.
Pressure hole d
Figure 5.3.a) Pressure hole locations with respect to the fracture mid-planes; b) 3-D view of the 10' drill hole (only wellbore pressure holes are shown) and c )
magnified cross-sections showing details of 20' model ( x 3 . O ) and of pressure hole construction with tube adopter installed ( x 5.0) (INSET). Note the enlargement of
C1
the wellbore sections to accommodate the PVC pipe extension.
set-up paramem for each nm series.
RUN NO.+ 1 2 3 4
h: -0.322 0.151 -0.947 -1.382 -1.599 -0.918 Re,,: 6478 4057 8753 10056 10649 8646 R%B : 308 193 417 479 507 412
Q: 0.247 0.304 0.355 0,151 0.277 Om176 T: 11.22 11.21 11-17 11.16 11.15 11.14 h: -2.037 -3.077 -4.170 -0.650 -2.568 -0.966
Re,,: 4907 6027 7038 2940 5491 3489
-6 : 234 287 335 142 261 166
Table 5.1. amt. !d
RUNNO.+ 1 2 3 4 5 6 7 8 SEZWP I _ _ _ - - - - = =
Q: 0,383 0.498 0.600 0.661 0.346 8 : 90'
Q: 0.193 0.312 0.366 0.537 0.601 0.664 0.375 0.451 T: 14-54 14.52 14.49 14.46 14.43 14.39 14.39 14.39 h$ 0.409 0.342 0.306 0.171 0.111 0.049 0.300 0.242
Re,,: 1089 1762 2068 3027 3385 3739 2108 2540
R%B : 167 271 318 465 520 575 324 390
T: 10.91 11.06 11.01 10.94 10.95 : 0.439 0.701 1.004 1.192 0.369
Re,,: 7548 9832 11842 13017 6817
: 0.183 0.327 0.401 0.262 0.072 0.091 0.U4 0.196 Re,,: 2086 3278 3804 2792 984 1235 1677 2211
Q: 0.182 0.283 0.323 T: 15.11 15.02 14.92 : 0.090 -0.190 -0.312
Re,,: 1042 1618 1845 %: 160 249 284
2b: 1.6 : 0.00
h,: 0.00
Q: 0.378 0.285 0.204 0.546 0.731 0.587 T: 15.29 15.12 15.13 15.14 15.10 15.06 : 0.075 0.043 0.029 0.121 0.179 0.132
Re,,: 2178 1637 1171 3133 4193 3362 ReoB : 335 252 180 481 644 517
-6 : 359 468 564 620 325 -ON
1 -3 3 ) units: Q (lt/s) or (10 - m / s ) ; T ('C); h (m); 2b (m) or (10-~*m)
'1 Subscripts: w ( w e l w r e ) ; IB (inner ; OB (outer b a m d a q )
6 ANALYSIS AND DISCUSSION OF TEST RESULTS
6.1 Introduction
The second phase of the laboratory study involves critical
examination of the experimental data outlined in Table 5.1 and fully
listed in Appendix 8. In this phase the objectives are two-fold:
a) to assess the performance of the laboratory set-up; and
b) to fulfil the general objectives of the laboratory study, i .e. to
determine the agreement between measured head profiles and those
predicted by Equation 4.29, and the exit/entry loss coefficients in
this equation as a function of the intersection angle.
The chapter begins with a background section (Section 6.2) reviewing
available information relevant to analysis procedure and theory. The
results are presented and interpreted in separate sect ions for pumping
(Section 6.3) and injection (Section 6.4) tests. The results of the test
series are illustrated graphically in each section. Verification of the
laboratory set-up performance and the data quality is based on the pumping
test results from the 90° model (Section 6.3). The chapter concludes with
a brief sumnary of the significance o f the laboratory study (Section 6.5).
6.2 Background
Knowledge of flow processes and resultant head losses associated
with sudden changes in flow area and direction i s essential to the
interpretation of the data. Flow visualization experiments using
contracting/enlarging sections (JSME, 1988) provide direct evidence for
the scale and geometric variables of related processes. Numerous other
96
studies as compiled by, for example, Fried and Idelchik (1989) examine the
resultant head losses in conduits of various shapes. These one-dimensional
flow studies provide a basis for interpreting the data from
accelerating/decelerating flows. The following is aimed to concisely
introduce the relevant aspects of the present knowledge.
Fluid flowing through a sharp-edged entrance of an abruptly
contracting conduit separates from the walls and forms a compressed jet
(Figure 6 . l . a ) . The acceleration of fluid mass, resulting from resistance
to the sharp turn in flow direction, ends at the vena-contracta where
effective flow area is minimum. After this point, the jet quickly expands
to f i l l the conduit. Strong deceleration results in an adverse pressure
gradient (i.e. increasing in the flow direction) which favours formation
of unsteady eddies causing dissipation of mechanical f l o w energy (Vennard
and Street, 1982). The entry process is completed as the eddies decay
downstream and the velocity profile is fully established. The distance
along which the whole process takes place is called the entry length
(Figure 6.l.a).
The entry process in the case of acute intersections (Figure 6.1.b)
is notably different than observed i n orthogonal entry sections. The
separation zone is asynmetric, irregularly enlarged and very unsteady, and
the entry length is longer. These visual differences due to different
intersection angles are also manifested in the magnitudes of head losses,
usually modelled by a dimensionless loss coefficient (Fried and Idelchik,
1989)
where
97
A p : static pressure change between end points of the entry
length, and - V: average velocity of fully established flow.
This expression is equally valid for the calculation of exit as well as
entry loss coefficients. Variation of this coefficient with the
intersection angle i s summarized in Figure 6.2. for Reynolds numbers
(Equation 2.19) greater than 10'.
Ward-Smith (1980) reviewing the numerical solutions of the Navier-
Stokes equation (Equation 2.1) o f co-axial flow into infinite parallel
plates recommended an entry loss coefficient of f ,=0 .662, and an entry
length, L, given by
L=0.022 Re (2b) ( 6 - 2 )
for R e > l 0 ' . Based on similar empirical expressions, Rissler (1978)
derived loss coefficients of e,=o .7ll and e , = O .415 for laminar and
turbulent co-axial flows, respectively, into parallel plate contractions.
Flow adjustment to sudden co-axial enlargements involves similar
processes: flow separation, eddy formation, expansion of jet and re-
attachment (Figure 6.3.a). The resultant head losses are again related to
incomplete pressure recovery. The exit loss coefficient, ex, for uniform
velocity prof i les is usual ly given by the Borda-Carnot relation (Ward-
Smith, 1980)
98
where 4 and A, are flow areas at the up- and down-stream faces of the
enlargement, respectively. In accordance with this relation, Fried and
Idelchik (1989) showed from the experimental literature that the exit loss
coefficient is not sensitive to the intersection angle when the velocity
of the passing stream, v,, in the enlarged section is much less than the
exit velocity, v, (Figure 6.3.b). In fracture-wellbore systems, this
condition likely often holds if one identifies 4 and A, with the exit and
wellbore areas, and V, and v, with the exit and wellbore velocities. For
experimental analysis, however, the magnitude of exit losses will be
investigated using Equation 6.1.
Accounting f o r the effects of variable flow section
The Poiseuille law (Equation 2.10) governs one-dimensional laminar
flow through uniform parallel-plate conduits where the velocity profile is
invariably parabolic. The kinetic energy correction factor (KECF)
(Equation 4.2) for such flows has a constant value of 1.54 (Vennard and
Street, 1982). In convergent/divergent flows, this value is relevant at
some distance from the wellbore depending on the flow rate. Closer to the
we1 lbore where the velocity becomes less parabol ic (and more uniform), the
KECF approaches unity. The knowledge of the KECF as a function of the
distance from the wellbore face is therefore necessary in order to
differentiate between the contributions of the frictional head loss and
the variable velocity head to the observed hydraulic head changes
(Equation 4.4).
99
Murphy (1979) compared a numerical solution o f the Navier-Stokes
equation (Equation 2.1) with its approximate analytical solutions which
assume constant KECFs of 1.54 and 1.2 at all radii. Since the velocity
head at large radii is negligible, the latter solution provides a better
overall fit to the numerical predictions. Therefore the constant KECF o f
1.2, which was also adopted by Baker (l955), Louis and Maini (1970) and
Rissler (1978), will be used in the analyses o f pressure data in the
linear domain.
6.3 Puqing tests
Inspection of the experimental data (Appendix B ) revealed that: a)
exit losses are not Reynolds number dependent in the test range; and b) no
systematic changes in percent discrepancy between measured and predicted
pressure heads exists between the runs of any pumping test series. General
characteristics of the data can therefore be exemplified by any one run
from each series. The runs with most equal flow rates were selected for
the graphical presentation of the data from corresponding series of each
model (Figures 6.4 to 6.9). This selection allows the most direct
comparison of the measured pressure profiles as a function o f the
intersection angle.
The graphs (Figures 6.4 to 6.9) consist of the direct pressure head
measurements along all the pressure hole arrays (Figure 5.3.a) and the
predicted pressure heads calculated from Equation 4.29 (with the total
head loss term expressed by Equation 4.4). Equivalent radius (Equation
A.15) adapted as abscissa is a useful concept which: a) allows a scaled
100
comparison between sizes of exit areas, and b) offers an alternative
visualization of an equivalent radial pressure distribution as a function
of wellbore radius.
Despite a fastidious fabrication and experimentation procedure, the
development of uncertainty in the results i s inevitable and can be
attributed to interacting uncertainties associated with all the measured
variables, dimensions and geometries (Kline and McClintock, 1953). The
only measure of the extent o f the resultant uncertainty can be obtained by
comparing the 90' model data with the radial flow predictions established
by earlier experiments (Rissler, 1978; Murphy, 1979). Measuring
essentially the same pressure profiles along all pressure hole arrays of
the 90' model and excellent agreement of these with the predicted profiles
suggest that the laboratory set-up operated as designed. For a given run,
discrepancy (between measured and predicted pressures) at any point tends
to be proportional to the pressure gradient, clearly because of great
sensitivity o f pressure to aperture variations. Percent discrepancy over
the full scale of the pressure difference is generally less than five
percent for all runs.
Laminar flow predictions were proven valid despite high Reynolds
number range attained particularly in 90' model tests (Table 5.1). As a
result of convergent radial air flow experiments in a similar smooth
model, Murphy (1979) also noted that measured profiles are consistent with
the laminar predictions for Reynolds numbers ranging from 210 to 20700.
Convergent, radial water flow through an open, rough fracture, however,
reveals a considerably 1 imited but still an extended range for laminar
101
flows up to Reynolds numbers of 4000 to 8000 (Baker, 1955). Murphy (1979)
suggested that agreement at low Reynolds numbers establishes the validity
of the experimental procedures and measurements. The agreement above the
critical Reynolds number o f 2300 (Figure 2 4 , marking onset of the
turbulence in one-dimensional flow, is attributed to the stabilizing role
of positive pressure gradients on boundary layers in accelerated flow
(Schlichting, 1979). Accordingly, during pumping tests, the laminar-
turbulent transit ion in fractures characterized by k / D h r 0.033 may be
expected to occur at Reynolds numbers higher than 2300.
Variation in pumping pressure head as a function o f the intersection
angle is we1 1 predicted by the mathematical model. Minor entry losses frorn
the model tank into the fracture are satisfactorily estimated using
Equation 6.1 with t,=o.662. As can be found from Equation 6.2, flow is
likely fully established before reaching the first pressure hole from the
outer boundary. The percent discrepancy or the qua1 ity of predictions for
the pumping test runs with the closure aperture (FSgures 6.7 to 6.9) are
not affected from the offset o f the wellbore axis in the acute models.
The review in the preceding section indicates that exit losses are
independent of the intersection angle in one-dimensional flow. The exit
loss coefficient is substantiated to be approximately unity for
accelerating flow. The fact that the wellbore axis is oblique to the
fracture plane in the acute system models did not seem to affect the
magnitude of the exit losses. This is probably because, although
streamlines approaching the wellbore frorn different directions undergo
bending in varying degrees, the overall loss is balanced out.
102
The pressures measured in the downstream section of the we1 lbore
were slightly below that measured at the we1 1 bottom. Since the span of
the monitored section was limited to two and four wellbore diameters for
the 20' and 10' models, respectively, it was not possible to differentiate
between entry effects and established flow losses in the wellbore.
However, the data suggest no noticeable variation in wellbore losses due
to changes in the intersection angle.
6.4 Injection tests
The injection pressure heads from the selected runs of each model
are plotted on successive graphs (Figures 6.10 to 6.15) using different
scales to emphasize variations along different pressure hole arrays.
Similar discharge rates of the runs representing the group of test series
for each aperture value facilitate direct comparison of the results as a
function o f the intersection angle. The distinct pressure patterns
recognized for each o f these runs are common characteristics of their
series. Uncertainty in pressure head measurements stated in the previous
section also applies to the injection tests.
The pressure head profiles predicted by Equation 4.29 and the entry
loss coefficients are also illustrated on the graphs. The coefficients
were calculated in terms of the entry velocity although associated losses
vary continuously over the entry length. This calculation procedure causes
the injection head to excessively drop at the entrance to the fracture to
a level such that the total head at the outer boundary approximates zero.
It was assumed here that the entry losses reach maximum within the model
103
boundary.
Pressure measurements near the wellbore boundary provide direct
evidence for the vena-contracta formation and its magnitude (Figures 6.10
to 6.15). Note, however, that two of the holes in the 20' model which are
located nearest to the boundary and along the major axis of the inlet
section (hole no. 1 and 9 in Figure B.l) are largely exposed to the
wellbore pressure regfme due to the obliquity of the inlet section
(Figures 6.11 and 6.14).
The pressure head profiles measured along different arrays are
variable for the acute models. This is most distinguishable for the 10'
model tests exhjbiting two distinct grouping o f the measurements: along
the arrays parallel to the minor axis of the inlet (involving 90' bending
of stream1 ines) and the other arrays (Figures 6.12 and 6.15). For the runs
simulating opening of the fracture (Figures 6.13 to 6.15). the profiles
follow similar patterns. Consequently, straight streamline visualization
cannot be strictly true because of non-axismetric pressure distribution
observed during injection tests i n acute models.
The entry loss coefficients for all models form a tight cluster
between f,=0.65-0.71 and e,=o.65-0.75 for the fabrication (2b=l.l
mm) and opening (2b=1.6 m) aperture series, respectively. Note that, as
the intersection angle decreases and/or aperture increases, the inlet area
increases and hence the ratio of the velocity in the wellbore to that at
the inlet increases. A slight increase in the coefficients from the 90' to
10' models and from the fabrication to opening apertures is therefore
consistent with the results of one-dimensional flow experiments (Figure
104
6.2.Inset). It follows that the choice of the entry loss coefficients
should be based on the ratio of the we1 lbore to inlet areas. The values of
the coefficients calculated for Reynolds numbers up to 12000 span the
range indicated by Rissler (1978) and Ward-Smith (1980) for one-
dimensional flow between parallel plates. The coefficients of all test
series appear to be independent of Reynolds number as in one-dimensional
flow.
The mathematical model (Equation 4.29) incorporating the empirical
entry loss coefficients is capable of reliably estimating one of the three
basic test parameters (fracture aperture, injection head and flow rate)
given the other two. Nevertheless, it hardly reproduces the in-fracture
injection pressure distribution because of vena contracta formation and
decay in all models, and small magnitude directional variations in the
acute models. The results clearly demonstrate prevalence of the entry
processes in determining pressure distribution around the wellbore where
the turbulent regime i s also most Influential. This situation is treated
in terms o f an equivalent system of two superimposed independent
processes. However, all these have little practical consequence because
what needs to be measured/predicted is the total head at the boundaries
rather than the actual energy transformation path within the fracture.
6.5 Sumry
The Reynolds numbers calculated at the wellbore face (Table 5.1)
suggest that the findings of this experimental study apply to a broad
range o f practical situations. With this incentive in mind, a better
105
understanding o f the near we1 1 flow mechanisms taking place during pumping
and injection tests has been established for the acute systems. The semi-
analytical model (Equation 4.29) as a predictive tool has been verified
and refined as a result of the laboratory study. The observations have
been comprehensively discussed to provide a guidance for the deductions.
jj..
Measurement location map
- Calculated projite
Stalk water level - - - - - - in model tank
b F~aolure mid-plum
PUMPING SERIEi
RUN NU. 8
Equivalent radius (m)
Figure 6 . 6 . Pumping pressure head vs. logarithm of equivalent radius: B=1o0; 2b=l.l nm; Run no. 8 (Table 5.1).
llemrement Location map
fl Calculated profile
Static water level I
in model tank
Fracture mid - p lane
PUMPING SERIES
RUN NO. 3
Equivalent radius (m)
C1 Figure 6 . 7 . Pumping pressure head vs. logarithm of equivalent radius: 8=90' ; 2b=O. 6 CU
mm; R u n no. 3 (Table 5.1).
Staliu water h e 1 - - - - - - in mods1 tank
+ Fmoture mid-plant
PUMPING SERIE2
Equivalent ~adius (m)
w Figure 6.9. Pumping pressure head vs. logarithm of equivalent radius: 8=10' ; 2b=0.6 - a
mm; Run no. 3 (Table 5.1).
Measurement tocaZian map
/ Calculated profile
+ Fracture mid-plane
INJECTION SERIES
RUN NO. 6
Equivalent radius (m)
Figure 6.10. Injection pressure head vs. logarithm of equivalent radius: B=90° ; + ~ l r
VI 2bsl.l mm; Run no. 6 (Table 5.1).
7 S-Y AND CONCLUSIONS
The problem of well test interpretation in acute systems has been
investigated both theoretically and experimentally, and presented
following a unifying and comprehensive approach. This investigation : a)
establishes a basic understanding of the near wetlbore flow mechanism in
acute systems; b) formalizes the intersection angle dependent variations
in the streamline pattern and hence in pressure distribution and observed
response; and c) provides mathematical tools to predict these variations.
In the following, the sumnary of the underlying work and the conclusions
(including their implications and reco~mendations) are presented
separately for each component.
The theoretical component o f this study involved:
a) the derivation of the governing differential equation of flow in
fractures o f acute systems,
b) the introduction of analytical models for constant-flux tests
under transient and steady-state conditions,
C) the formulation of the streamline-equipotential network created
by the injection/pumping through acute systems under initially non-
uniform heads, and
d) the development o f a general, semi-analytical model accounting
for the roughness, turbulence and intersection effects in
interpreting single-well constant-flux tests.
The main conclusions are enumerated as follows:
1. The early period during which the system intersection angle dominates
122
the response is extended by a factor of (eP/%,l2 as this angle becomes
smaller. Ignoring the intersection angle in the evaluation of data from
the early period w i 1 1 therefore lead to an overestimate o f transmissivity.
2. The zone o f influence is essentially independent of the intersection
angle. Thus, the biased response i n reflecting the properties of the near
wellbore area (due to logarithmic pressure distribution) is not further
complicated due to non-orthogonality of the fracture-wellbore
intersection. This implies that single-fracture permeability tests can be
utilized in a network model without any reservations.
3. Interpretation of steady, single-well constant-flu x tests b j f assuming
an orthogonal system will produce overestimated aperture and hence
hydraulic conductivity values.
4. In developing production strategies, it should be considered that: a)
well losses can be minimized naturally by producing a system with the
lowest possible intersection angle: and b) efficiency of and interference
distance between producing wells largely depends on the intersection angle
of the effective fracture(s) in the producing intervals.
5. A single wellbore may be adequate to obtain anisotropic point
permeability in a fractured rock mass. This may result i n a more
homogenous sampling and/or substantial saving in drilling costs.
123
6. The predictions of any single-well test model based on an orthogonal
system assumption can be refined by simply substituting the actual
wellbore radius w i t h the equivalent radius, e .
7. The single-fracture, constant-f lux test models developed in this study
can be readily extended to the interpretation of tests conducted in long
wellbore intervals with multiple fractures by weighing the intersection
angle of each subsystem with reference to the aperture distribution.
A considerable effort has been made to undertake a laboratory study:
a) to verify the formation of the idealized stream1 ine pattern and examine
effects of likely interactions at the acute intersections particularly
during injection tests; and b) to quantify the exit/entry loss
coefficients as a function of the intersection angle. The original
experimental set-up designed to carry out this investigation includes
three distinct fracture-wellbore system models with g o 0 , 20' and 10'
intersection angles. The laboratory programme involved testing these
models for three different ( i . e . fabrication, 2b=l.l mm; closure, 2b=0.6
nm; opening, 2b=1.6 nm) apertures under steady, constant-f lux, injection
and pumping conditions.
The overall experimental set-up successfully simulated the
conceptual testing environment which the mathematical model is expected to
reproduce. It was therefore possible: a) to test the predicted pressure
distribution and response; b) to interpret the deviations in terms o f near
wellbore flow mechanisms; and c) to derive the entry and exit loss
coefficients. The main conclusions reached are:
1. The mathematical models were successful in predicting the response
observed in single-well, pumping tests and they are appropriate for
interpreting the field test data. The quality o f predictions are not
influenced from the offset of the we1 lbore axis due to fracture closure-
2. The exit loss coefficient is virtually unchanged from unity and not
affected by fracture closure.
3. The streamline pattern during injection tests, at both fabrication and
opening apertures, however, does not form exactly as conceptualized. The
reason for this i s the dependence of the entry length, position and
intensity of vena-contracta, and subsequently the distribution of the
entry losses, on the streamline position with respect to intake section.
Having recognized this limitation, the developed mathematical models still
provide the best alternative in interpreting single-well injection tests
in acute systems.
4. Entry loss coefficient i s not very sensitive to the intersection angle
and can be roughly taken as 0.65 for all angles, probably less than about
5'. In field applications, this coefficient may slightly change for
different ratios of wellbore to inlet areas due to differences in wellbore
radius and/or apertures.
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AN EXPRESSION FOR THE PERIMETER OF AN EQUIPOTENTIAL SURFACE IN A
COWVERGENT/DIVERGENT F L W FIELD C(IWSED OF STRAIGHT SlREAllLINES mWUYU
TO THE ELLIPTICAL INNER BOUNDARY
The parametric equations o f an ellipse (Figure A. 1) are
where a: semi-major axis,
b: semi-minor axis, and
0: position angle.
Equation A. 1 defines the position vector describing the ellipse by
Taking the derivative of the position vector B produces a vector ?
tangent to the ellipse,
A vector n j normal to this tangent vector P can be found from the
orthogonality condition
f i (0) =; b cose +f a sine
The normal vector a can be assigned any length 1 to obtain
Hence, the position vector for an equipotential surface at any normal
distance 1 from the ellipse is defined by (Figure A.1 )
~ c e ) = ~ c e ) + ~ c e ) (A-7)
which enables writing the parametric equations of an equipotential surface
in the same plane as
Any expression f o r the perimeter r o f such an equipotential surface
needs to satisfy the general arc length equation for closed curves
136
(A. 10)
Differentiating Equations A.8 and A.9, and, substituting these in Equation
A.10, the perimeter I' expression is established as
Referring to Equation A.10, the perimeter of an ellipse l', i s
readily obtained as
The expression for the perimeter I' (Equation A . l l ) then reduces to
(A. 13)
Solving for the integral in the second term, Equation A.13 takes on a
compact form
(A. 14)
137
Rearranging Equation A. 14 results in a more famil iar format, the perimeter
expression for an equivalent circle,
(A. 15)
where the term ( a + l ) may be considered as the equivalent radius.
The perimeter of an ellipse r, (Equation A.12) is usually given by
(A. 16)
where E ( K , x / 2 ) is the complete elliptical integral given by (Tuma, 1987)
The values of this function are tabulated in most mathematical handbooks
(e.g. Tuma, 1987).
An interesting point i s that Equation A . 1 6 is a general result,
valid regardless of the geometry of the inner flow boundary. For
applications with other geometries the term r, should be replaced with the
corresponding perimeter expression.
Figure A.1 . Pictorial definition of t h e terms used in the der ivat ion.
IN-FRACTURE PRESSURE HEAD MEASUREMENTS
1. The pumping and injection series are tabulated separately for
g o 0 , 20' and 10' models.
2. All pressure head values are given i n meters.
3. Measurement hole locations are illustrated in Figure B . I .
PUMPING SERIES : 6: 90' Lb: 1.1 mrn
PUMPlNG SERIES :
Run no. L - 7 3. 4 5 6
Hole
') For the hole locations, refer to Figure A.2.
') For the pressure head vs. logarithmic distance grapbr of the marked runs, refer to Figurer 6.4 to 6.15
PUIMPING SERIES : B:20e 2b: l . lmm
Rum no. 1 - 7 3 4 5 6' 7
Hole QrOSlOlh Qr0.757f.h Q0.6Bllfs Q0.182Vr G O m l h Q.0-443k QOmlh
no- T: 1458 'C T: 1435 T T: LA59 "C T: 1357 'Y= T: 1184°C T: U.68 'C T: 14-10 "C
P U M P I N G SERIES :
Rua no. 1' 2
Hole Q03631h Q:029Vs
PUMPlNG SERIES : 8: 10' 2b: 1.1 m m
Run no. 1 2 3 4 5 6 7 8
P U M P I N G SERIES :
Run no. 1 - 7 3
INJECTION SERIES : 0: 90" Lb: 1.1 m m
no. 16.21 'C T: 15.69% T: L5.6S"C T: I580 "C T: l559T. T: 15-61 "C
1 -0.308 -0.744 - 1.099 - 1.653 - 1.956 -2.370
Runno. 1 - 7 3' 4 ---- -- 5 -.
Hole QOJ83 lh Q.. 0.198 1h Q 0.6QOkk Q 0.641 lh Q 0.346 th
no. T: 10-91 "C T: 1L06 'C T: 1LO1 'C T: 1034 T T: LW 'C
1 -0.586 -0.936 - 1.328 - 1.608 -0.183
7 - - 0.004 - 0.053 - 0.095 - 0.142 0.0 LO
3 0.027 0.019 0.012 0.007 0.03 1
tNJECTION SERIES : 6: 20' 1b: 1.1 m m
INJECTION SERIES : - -
[NJECTION SERIES : 6: 10' 2b: 1.1 mm
INJECTION SERIES : 8: 10' tb: 1.6 mm
APPLIED 1 IMAGE. lnc = 1653 East Main Street - -. - Rochester. NY 14609 USA -- -- - - Phone: 71 61482-0300 -- -- - - F w 7161288-5989